algebra math notes abstract algebra
TRANSCRIPT
Algebra Math Notes โข Study Guide
Abstract Algebra
Table of Contents Rings ................................................................................................................................................... 3
Rings ............................................................................................................................................... 3
Subrings and Ideals ......................................................................................................................... 4
Homomorphisms .............................................................................................................................. 4
Fraction Fields ................................................................................................................................. 5
Integers ............................................................................................................................................ 5
Factoring ............................................................................................................................................. 7
Factoring .......................................................................................................................................... 7
UFDs, PIDs, and Euclidean Domains .............................................................................................. 7
Algebraic Integers ............................................................................................................................ 8
Quadratic Rings ............................................................................................................................... 8
Gaussian Integers ............................................................................................................................ 9
Factoring Ideals ............................................................................................................................... 9
Ideal Classes ................................................................................................................................. 10
Real Quadratic Rings ..................................................................................................................... 11
Polynomials ....................................................................................................................................... 12
Polynomials ................................................................................................................................... 12
โ[๐] ............................................................................................................................................... 12
Irreducible Polynomials .................................................................................................................. 13
Cyclotomic Polynomials ................................................................................................................. 14
Varieties ......................................................................................................................................... 14
Nullstellensatz ................................................................................................................................ 15
Fields................................................................................................................................................. 17
Fundamental Theorem of Algebra .................................................................................................. 17
Algebraic Elements ........................................................................................................................ 17
Degree of a Field Extension ........................................................................................................... 17
Finite Fields ................................................................................................................................... 19
Modules ............................................................................................................................................. 21
Modules ......................................................................................................................................... 21
Structure Theorem ......................................................................................................................... 22
Noetherian and Artinian Rings ....................................................................................................... 23
Application 1: Abelian Groups ........................................................................................................ 23
Application 2: Linear Operators ...................................................................................................... 24
Polynomial Rings in Several Variables ........................................................................................... 25
Tensor Products ............................................................................................................................ 25
Galois Theory .................................................................................................................................... 27
Symmetric Polynomials .................................................................................................................. 27
Discriminant ................................................................................................................................... 27
Galois Group .................................................................................................................................. 27
Fixed Fields ................................................................................................................................... 28
Galois Extensions and Splitting Fields ........................................................................................... 28
Fundamental Theorem ................................................................................................................... 29
Roots of Unity ................................................................................................................................ 29
Cubic Equations ............................................................................................................................. 30
Quartic Equations .......................................................................................................................... 30
Quintic Equations and the Impossibility Theorem ........................................................................... 31
Transcendence Theory .................................................................................................................. 32
Algebras ............................................................................................................................................ 33
Division Algebras ........................................................................................................................... 33
References ........................................................................................................................................ 34
1 Rings 1-1 Rings
A ring ๐ is a set with the operations of addition (+) and multiplication (ร) satisfying:
1. R is an abelian group ๐ + under addition. 2. Multiplication is associative. In a commutative ring, multiplication is commutative. A
ring with identity has a multiplicative identity 1. 3. Distributive law:
๐ ๐ + ๐ = ๐๐ + ๐๐, ๐ + ๐ ๐ = ๐๐ + ๐๐ A left/ right zero divisor is an element ๐ โ ๐ so that there exists ๐ โ 0 with ๐๐ = 0, ๐๐ = 0, respectively. A commutative ring without zero divisors is an integral domain. A ring is a
division ring if ๐ โ {0} is a multiplicative group under multiplication (inverses exist). A division ring is a field if the multiplicative group is abelian.
Ex. of Rings: โ, ๐ [๐ฅ] (the ring of polynomials in x, where R is a ring), ๐ [๐ฅ1, โฆ , ๐ฅ๐ ]
Note: A field is a UFD, though not a very interesting one. Basic properties:
1. ๐0 = 0๐ = 0 2. ๐ โ๐ = โ๐ ๐ = โ(๐๐) 3. The multiplicative identity 1 is unique (if it exists). 4. A multiplicative inverse is unique if it exists.
5. If R is not the zero ring {0}, then 1 โ 0.
Group G
Abelian group
Ring R
Commutative ring
Ring with identity
Integral domain Division ring
Field F
Principal ideal domain (PID)
Euclidean domain
Unique factorization domain (UFD)
6. If R is an integral domain, the cancellation law holds. Conversely, if the left (right) cancellation law holds, then R has no left (right) zero divisors.
7. A field is a division ring, and any finite integral domain is a field. From here on rings are assumed to be commutative with identity unless otherwise specified. A unit is an element with a multiplicative inverse.
The characteristic of a ring is the smallest ๐ > 0 so that 1 + โฏ + 1 ๐
= 0. If this is never true,
the characteristic is 0.
1-2 Subrings and Ideals A subring is a subset of a ring that is closed under addition and multiplication. An ideal I is a subset of a ring satisfying:
1. I is a subgroup of ๐ +. 2. If ๐ โ ๐ผ, ๐ โ ๐ then ๐๐ โ ๐ผ.
Every linear combination of elements ๐ ๐ โ ๐ผ with coefficients ๐๐ โ ๐ is in I. The principal ideal ๐ = ๐๐ generated by ๐ is the ideal of multiples of ๐.
The smallest ideal generated by ๐1, โฆ , ๐๐ is ๐1, ๐2 , โฆ , ๐๐ = ๐1๐1 + โฏ + ๐๐๐๐ ๐๐ โ ๐
Ideals and fields:
1. The only ideals of a field are the zero ideal and the unit ideal. 2. A ring with exactly two ideals is a field. 3. Every homomorphism from a field to a nonzero ring is injective.
A principal ideal domain (PID) is an integral domain where every ideal is principal.
Ex. โ is a PID. If F is a field, ๐น[๐ฅ] (polynomials in x with coefficients in F) is a PID: all polynomials in the ideal are a multiple of the unique monic polynomial of lowest degree in the ideal. A maximal ideal of R is an ideal strictly contained in R that is not contained in any other ideal.
1-3 Homomorphisms A ring homomorphism ๐: ๐ โ ๐ โฒ is compatible with addition and multiplication:
1. ๐ ๐ + ๐ = ๐ ๐ + ๐(๐) 2. ๐ ๐๐ = ๐ ๐ ๐(๐)
3. ๐ 1 = 1 An isomorphism is a bijective homomorphism and an automorphism is an isomorphism
from R to itself. The kernel is {๐ โ ๐ |๐ ๐ = 0}, and it is an ideal.
1-4 Quotient and Product Rings (11.4,6) If I is an ideal ๐ /๐ผ is the quotient ring. The canonical map ๐: ๐ โ ๐ /๐ผ sending ๐ โ ๐ + ๐ผ is
a ring homomorphism that sends each element to its residue. If ๐ผ = (๐1, โฆ , ๐๐ ), the quotient
ring is obtained by โkillingโ the ๐๐ , i.e. imposing the relations ๐1, โฆ , ๐๐ = 0.
Mapping Property: Let ๐: ๐ โ ๐ โฒ be a ring homomorphism with kernel K and let ๐ผ โ ๐พ be an
ideal. Let ๐: ๐ โ ๐ /๐ผ = ๐ be the canonical map. There is a unique homomorphism ๐ : ๐ โ ๐ โฒ so that ๐ ๐ = ๐.
First Isomorphism Theorem: If f is surjective and ๐ผ = ๐พ then ๐ is an automorphism. Correspondence Theorem: If f is surjective with kernel K,
There is a bijective correspondence between ideals of ๐ and ideals of ๐ โฒ that contain
๐พ. A subgroup of G containing K is associated with its image.
If ๐ผ corresponds to ๐ผโฒ, ๐ /๐ผ โ ๐ โฒ/๐ผโฒ. Corollary: Introducing relations one at a time (killing ๐, then ๐ ) or together (killing ๐, ๐)
gives the same ring. Useful in polynomial rings.
The product ring ๐ ร ๐ โฒ is the product set with componentwise addition and multiplication. 1. The additive identity is (0,0) and the multiplicative identity is (1,1).
2. The projections ๐ ๐ฅ, ๐ฅโฒ = ๐ฅ, ๐ โฒ ๐ฅ, ๐ฅโฒ = ๐ฅโฒ are ring homomorphisms to ๐ , ๐ โฒ. 3. 1,0 , (0,1) are idempotent elements- elements such that ๐2 = 1.
Let e be an idempotent element of a ring S.
1. ๐โฒ = 1 โ ๐ is idempotent, and ๐๐โฒ = 0. 2. ๐๐ is a ring (but not a subring of S unless e=1) with identity e, and multiplication by e
is a ring homomorphism ๐ โ ๐๐. 3. ๐ โ ๐๐ ร ๐โฒ๐.
1-5 Fraction Fields Every integral domain can be embedded as a subring of its fraction field F. Its elements
are ๐
๐ ๐, ๐ โ ๐ , ๐ โ 0 , where
๐1
๐1=
๐2
๐2โ ๐1๐2 = ๐2๐1. Addition and multiplication are defined
as in arithmetic: ๐
๐+
๐
๐=
๐๐ +๐๐
๐๐,๐
๐
๐
๐=
๐๐
๐๐, and
๐
1= ๐. The field of fractions of the polynomial
ring ๐พ[๐ฅ], K a field, is the field of rational functions ๐พ(๐ฅ). Mapping Property:
Let R be an integral domain with field of fractions F, and let ๐: ๐ โ ๐นโฒ be an injective homomorphism from ๐ to a field ๐นโ. ๐ can be extended uniquely to an injective
homomorphism ฮฆ: ๐น โ ๐นโฒ by letting ฮฆ ๐
๐ = ๐ ๐ ๐ ๐ โ1.
1-6 Integers
Peanoโs Axioms for โ: 1. 1 โ โ 2. Successor function: The map ๐: โ โ โ sends an integer to the next integer ๐ฅโฒ =
๐(๐ฅ); ๐ 1 = 2, ๐ 2 = 3 โฆ ๐ is injective, and ๐ ๐ โ 1 for any ๐ (i.e. 1 is the first element).
3. Induction axiom: Let S be a subset of โ having the following properties: a. 1 โ ๐
b. If ๐ โ ๐ then ๐โฒ โ ๐. Then ๐ = โ. (Counting runs through all the natural numbers.)
๐ ๐ โฒ
๐ = ๐ /๐ผ
๐
๐ ๐
Inductive/ Recursive Definition: By induction, if an object ๐ถ1 is defined, and a rule for determining ๐ถ๐ โฒ = ๐ถ๐+1 from ๐ถ๐ is given, then the sequence ๐ถ๐ is determined uniquely.
Recursive definition ofโฆ
1. Addition: ๐ + 1 = ๐โฒ, ๐ + ๐โฒ = (๐ + ๐)โฒ (i.e. ๐ + ๐ + 1 = ๐ + ๐ + 1)
2. Multiplication: ๐ ร 1 = ๐, ๐ ร ๐โฒ = ๐ ร ๐ + ๐ (i.e. ๐ ร ๐ + 1 = ๐ ร ๐ + ๐) From these, the associative, distributive, and distributive laws can be verified. (โPeano
playingโ) The integers can be developed from โ by introducing an additive inverse for every element.
2 Factoring 2-1 Factoring
Vocabulary
u is a unit if ๐ข has a multiplicative inverse in R
๐ข = (1)
a divides b if ๐ = ๐๐ for some ๐ โ ๐ ๐ โ (๐)
a properly divides b if ๐ = ๐๐ for some ๐ โ ๐ and neither ๐ nor ๐ are units
๐ โ ๐ โ 1
a and b are relatively prime
๐ and ๐ have no common divisor except units.
๐ โฉ ๐ = 1
a and b are associates if each divides the other, or if
๐ = ๐ข๐, u a unit
๐ = ๐
a is irreducible if a is not a unit and has no proper divisor (its only divisors are units and associates)
๐ โ 1 , and there is no principal ideal (๐) such that ๐ โ ๐ โ (1).
p is a prime element if ๐ is not a unit, and whenever ๐ divides ๐๐, ๐ divides ๐ or ๐
divides ๐.
๐๐ โ ๐ โ ๐ โ ๐ or ๐ โ (๐)
Ex. Nonunique factorization: 2 โ 3 = 6 = 1 + โ5 1 โ โ5 in โ โ5 .
2-2 UFDs, PIDs, and Euclidean Domains A size function is a function ๐: ๐ โ โ. An integral domain is an Euclidean domain if there is a norm ๐ such that division with remainder is possible:
For any ๐, ๐ โ ๐ , ๐ โ 0, there exist ๐, ๐ โ ๐ so that ๐ = ๐๐ + ๐, and ๐ = 0 (in which case ๐ divides ๐) or ๐ ๐ < ๐(๐).
Examples:
1. For โ, ๐ ๐ = ๐ . 2. For ๐น[๐ฅ], ๐ ๐ = deg ๐.
3. For โ[๐], ๐ ๐ = ๐ 2. A characterization of the GCD: Let R be a UFD. A greatest common divisor ๐ of ๐, ๐ โ ๐
is ๐ such that If ๐|๐, ๐ then ๐|๐. If R is a PID, this is equivalent to ๐ ๐ = ๐ ๐ + ๐ ๐, and there exist ๐, ๐ โ ๐ so ๐ = ๐๐ + ๐๐ (Bezout). ๐, ๐ are relatively prime iff ๐ is a unit. An integral domain is a unique factorization domain (UFD) if factoring terminates (stopping when all elements are irreducible), and the factorization is unique up to order and multiplication by units. The following are equivalent:
1. Factoring terminates. 2. R is Noetherian: it does not contain an infinite strictly increasing chain of principal
ideals ๐1 โ ๐2 โ โฏ.
Class (Each includes the next)
Definition Reasons Properties
Ring
Integral domain
Ring with no zero divisors
Prime element irreducible
Unique factorization domain (UFD)
Integral domain where
Factoring terminates
Factorization unique or equivalently, every irreducible element is prime.
For equivalence: If every irreducible element prime, and there are two factorizations, an element on the left divides an element on the right; they are associates and can be canceled.
Irreducible element prime GCD exists (factor and look at common prime divisors)
Principal Ideal Domain (PID)
All ideals generated by one element
A PID is a UFD: 1- Irreducible element prime:
๐ ๐ = ๐ ๐ + ๐ ๐ โ Bezoutโs identity. Irreducible element
prime since ๐ ๐๐, ๐ โค ๐ โ 1 =๐ ๐ + ๐ก๐ โ ๐ ๐ = ๐ ๐๐ + ๐ก๐๐ 2- Factoring terminates: If ๐1 โ ๐2 โ โฏ then โ ๐๐ = (๐) is a principal
ideal; ๐ โ (๐๐ ) for some j.
Maximal ideals generated by irreducible elements
Euclidean domain
Integral domain with norm compatible with division with remainder
A Euclidean Domain is a UFD: The element of smallest size in an ideal generates it.
Euclidean algorithm works. (Given a, b,
write ๐ = ๐๐ + ๐, replace ๐ with ๐, ๐ with ๐.)
Ex. โ, โ ๐ , ๐น[๐ฅ] are UFDs.
2-3 Algebraic Integers An algebraic number is the root of an integer polynomial equation. It is an algebraic integer if it is the root of a monic integer polynomial equation. Equivalently, its irreducible
polynomial over โ is monic. A rational number is an algebraic integer iff it is an integer.
2-4 Quadratic Rings
โ ๐ is a quadratic number field when ๐ โ โ is not a square.
The algebraic integers in โ ๐ฟ , ๐ฟ = ๐, d squarefree, have the form ๐ผ = ๐ + ๐๐ฟ, where
If ๐ โก 2,3(mod 4) then a and b are integers.
If ๐ โก 1(mod 4) then ๐ and b are both integers or half-integers (๐ +1
2). In other words,
they have the form ๐ + ๐๐, ๐ =1
2 1 + ๐ ; ๐, ๐ โ โ.
The algebraic integers in โ ๐ form the ring of integers R in the field, โ[๐ฟ] or โ[๐]. Complex quadratic rings can be represented by a lattice in the complex plane.
The norm is defined by ๐ ๐ง = ๐ง ๐ง.
The norm is a multiplicative function, and ๐ ๐ง = ๐(๐ง ).
Hence ๐ฅ ๐ฆ โ ๐ ๐ฅ ๐ ๐ฆ , ๐ฅ|๐ฆ โ ๐ ๐ฅ |๐ฆ๐ฅ . An element is an unit iff its norm is 1.
If ๐ = โ3, โ2, โ1,2,3,5 then R is an Euclidean domain, and hence a UFD. The only other values of d where R is a complex UFD are -7, -11, -19, -43, -67, and -163.
d Units Is 2 prime?
-3 ยฑ1, ยฑ
1
2ยฑ
3
2
Yes
-2 1, โ1 No
-1 ยฑ1, ยฑ๐ No
2 Infinitely many No
3 Infinitely many No
5 Infinitely many Yes
All primes in R have norm p or ๐2. If the integer prime ๐ is odd andโฆ
๐ is a perfect square modulo p, then p is the product of 2 conjugate primes of norm p.
๐ is not a perfect square modulo p, then p is prime in R. Some Theorems
1. Chinese Remainder: Given pairwise coprime ๐๐ and any ๐๐ , there exists a unique number ๐ฅ โ ๐ modulo โ๐๐ such that ๐ฅ๐ โก ๐๐(mod ๐๐).
2. ๐ /๐๐ has ๐(๐) elements. It is a field iff ฯ is prime. (Pf. If N(ฯ)=p then 1,โฆp are the
distinct residues. If ๐(๐) = ๐2 then ๐ = ๐; take ๐ + ๐๐, 1 โค ๐, ๐ โค ๐. Use induction on
the number of prime factors of ฯ.) 3. Fermatโs Little Theorem: ๐๐ ๐ โก ๐(mod ๐).
4. Eulerโs Theorem: ๐๐ ๐ โก 1(mod ๐).
5. Totient: ๐ โ ๐๐๐๐ ๐
๐=1 = ๐ ๐ โ๐ ๐๐ โ1
๐ ๐๐ ๐๐=1 .
6. Wilsonโs Theorem: If ฯ is prime, the product of all nonzero residues modulo ฯ is
congruent to -1 modulo ฯ. 7. There are finitely many pairwise non-associated numbers with given norm.
2-5 Gaussian Integers โ[๐] is the ring of Gaussian integers.
1. If ๐ is a Gauss prime, then ๐๐ is an integer prime or the square of an integer prime.
2. Each integer prime ๐ is a Gauss prime or the product ๐๐ where ๐ is a Gauss prime. 3. Primes congruent to 3 modulo 4 are Gauss primes. (They are not the sum of two
squares.) 4. Primes congruent to 1 modulo 4, and 2, are the product of complex conjugate Gauss
primes. Pf. (3-4)
p is a Gauss prime iff (๐) is a maximal ideal in โ[๐], iff ๐ = โ ๐ (๐) = ๐ฝ๐ ๐ฅ (๐ฅ2 + 1) is a
field, iff ๐ฅ2 + 1 is irreducible (doesnโt have a zero) in ๐ฝ๐[๐ฅ]. -1 is a square modulo ๐ iff ๐ = 2
or ๐ โก 1 (mod 4).
2-6 Factoring Ideals
If A and B are ideals, then ๐ด๐ต = ๐๐๐๐๐ ๐๐ โ ๐ด, ๐๐ โ ๐ต . ๐ is the unit ideal since ๐ด๐ = ๐ด for any R.
Let R be a complex quadratic ring. Then ๐ด๐ด = ๐ = ๐๐ for some n; i.e. the product
is a principal ideal. (Pf. Let ๐, ๐ , (๐ , ๐ ) be lattice bases for A, B. Let
๐ = gcd(๐ ๐, ๐ ๐, ๐ ๐ + ๐ ๐); then (๐) โ ๐ด ๐ด. Show n divides the four generators
๐ ๐, ๐ ๐, ๐ ๐, ๐ ๐, by showing ๐ ๐
๐,๐ ๐
๐ are algebraic integers, so ๐ด ๐ด โ ๐ .)
Cancellation Law: Let A, B, C be nonzero ideals. ๐ด๐ต = ๐ด๐ถ iff ๐ต = ๐ถ. ๐ด๐ต โ ๐ด๐ถ โ ๐ต โ๐ถ.
๐ด|๐ต โ ๐ด โ ๐ต. Note that if A divides B, then A is โlargerโ than B- it contains more elements; upon multiplication, many of the elements disappear.
A prime ideal P satisfies any of the following equivalent conditions:
1. ๐ /๐ is an integral domain. 2. ๐ โ ๐ , and ๐๐ โ ๐ โ ๐ โ ๐ or ๐ โ ๐. 3. ๐ โ ๐ , and if A, B are ideals of R, ๐ด๐ต โ ๐ โ ๐ด โ ๐ or ๐ต โ ๐.
Any maximal ideal is prime. If P is prime and not zero, and ๐|๐ด๐ต, then ๐|๐ด or ๐|๐ต. Letting R be a complex quadratic ring, and B be a nonzero ideal,
1. B has finite index in R. 2. Finitely many ideals in R contain B. (Pf. Look at lattice) 3. B is in a maximal ideal.
4. B is prime iff it is maximal. (Pf. ๐ /๐ is a field.) Every proper ideal of a complex quadratic ring R factors uniquely into a product of prime ideals (up to ordering).1 (Pf. Use 2 and cancellation law.)
Warning: Most rings do not have unique ideal factorization since ๐ โ ๐ต โ ๐|๐ต. The complex quadratic ring R is a UFD iff it is a PID. (Pf. If P is prime, P contains a prime divisor ฯ of some element in P. Then ๐ = (๐).) Below, P is nonzero and prime, p is an integer prime, and ฯ is a Gauss prime.
1. If ๐ ๐ = (๐) then ๐ = ๐ or ๐2 for some p.
2. (๐) is a prime ideal (p โremains primeโ), or ๐ = ๐ ๐ (p splits; if ๐ = ๐ then p ramifies).
3. If ๐ โก 2,3(mod 4), then p generates a prime ideal iff d is not a square modulo p, iff ๐ฅ2 โ ๐ is irreducible in ๐ฝ๐[๐ฅ]. [Pf. by diagram]
4. If ๐ โก 1 mod 4 , then p generates a prime ideal iff ๐ฅ2 โ ๐ฅ +1โ๐
4 is irreducible in ๐ฝ๐[๐ฅ].
Random: For nonzero ideals A, B, C, ๐ต โ ๐ถ โ ๐ต: ๐ถ = ๐ด๐ต: ๐ด๐ถ . (Show for prime ideal A, split into 2 cases based on (2).)
2-7 Ideal Classes Ideals A and B are similar if ๐ต = ๐๐ด for some ๐ โ โ. (The lattices are similar and oriented the same.) Similarity classes of ideals are ideal classes; the ideal class of A is denoted by
โ๐ดโ. The class of the unit ideal โ๐ โ consists of the principal ideals. The ideal classes form the (abelian) class group C of R, with โ๐ดโโ๐ตโ = โ๐ด๐ตโ. (Note โ๐ดโโ1 =โ๐ด โ.) The class number |๐ถ| tells how โbadlyโ unique factorization of elements fails. Measuring the size of an ideal:
Norm: ๐ ๐ด = ๐, where (๐) = ๐ด ๐ด. Multiplicative function.
Index [๐ : ๐ด] of A in R.
ฮ(๐ด), the area of the parallelogram spanned by a lattice basis.
Minimal norm of nonzero elements of A. Relationships:
๐ ๐ด = ๐ : ๐ด =ฮ ๐ด
ฮ ๐
If ๐ โ ๐ด has minimal nonzero norm, ๐ ๐ โค ๐ ๐ด ๐, ๐ =
2 ๐
3, if ๐ โก 2,3 (mod 4)
๐
3, if ๐ โก 1 (mod 4)
. (Pf.
๐ ๐ โค2
3ฮ(๐ด))
1. Every ideal class contains an ideal A with norm ๐ ๐ด โค ๐. (Pf. Choose nonzero
1 A Dedekind domain is a Noetherian, integrally closed integral domain with every nonzero prime ideal maximal (such as
the complex quadratic rings). โIntegrally closedโ means that the ring contains all algebraic integers that are in its ring of fractions. Prime factorization of ideals holds in any Dedekind domain. See Algebraic Number Theory.
element with minimal norm, ๐ = ๐ด๐ถ for some C, ๐ ๐ถ โค ๐, โ๐ถโ = โ๐ด โ,โ๐ดโ = โ๐ถ โ.) 2. The class group C is generated by โ๐โ, for prime ideals P with prime norm ๐ โค ๐. (Pf.
Factor. Either ๐(๐) = ๐ or ๐ = (๐).) 3. The class group is finite. (Pf. There are at most 2 prime ideals with norms a given
integer since ๐ด๐ด = (๐) has unique factorization; use multiplicativity of norm.)
Computing the Class Group of ๐ = โ[ ๐] . 1. List the primes ๐ โค โ๐โ. 2. For each p, determine whether p splits in R by checking whether ๐ฅ2 โ ๐ ๐ โก
2,3mod4, ๐ฅ2โ๐ฅ+1โ๐4 ๐โก1mod4 are reducible.
3. If ๐ = ๐ด ๐ด splits in R, include โ๐ดโ in the list of generators. 4. Compute the norm of some small elements (with prime divisors in the list found
above), like ๐ + ๐ฟ. ๐ โ โ. Factor ๐(๐) to factor ๐ ๐ = ๐ ๐ ; match factors using
unique factorization. Note โ ๐ โ = โ ๐ โ = โ๐ โ = 1. As long as ๐ is not divisible by one of the prime factors of ๐(๐), this amounts to replacing each prime in the factorization
of ๐ ๐ by one of its corresponding ideals in (3) and setting equal to 1. Repeat until
there are enough relations to determine the group. [This works since if โ โ๐๐โ๐๐
๐ =1, ๐ ๐๐ = ๐๐ then there is an element ๐, ๐ ๐ = โ ๐๐
๐๐๐ .]
5. For the prime 2,
a. If ๐ โก 2,3(mod 4), 2 ramifies: 2 = ๐๐ . ๐ has order 2 for ๐ โ โ1, โ2.
b. If ๐ โก 2(mod 4), ๐ = 2, ๐ฟ . c. If ๐ โก 3 mod 4 , ๐ = 2,1 + ๐ฟ .
2-8 Real Quadratic Rings
Represent โ[ ๐] as a lattice in the plane by associating ๐ผ = ๐ + ๐ ๐ with
๐ข, ๐ฃ = (๐ + ๐ ๐, ๐ โ ๐ ๐). (The coordinates represent the two ways that ๐น[๐ฟ] can be
embedded in the real numbers, where ฮด is the abstract square root of d: ๐ฟ2 = ๐.)
The norm is ๐ ๐ = ๐2 โ ๐2๐ (sometimes the absolute value is used).
The units satisfy ๐ ๐ = 1; they lie on the hyperbola ๐ข๐ฃ = ยฑ1. The units form an infinite group in R. 2 proofs: 1. The Pellโs equation has infinitely many solutions.
2. Let ฮ be the determinant for the lattice of โ[ ๐] and ๐ท๐ = {(๐ข, ๐ฃ)|๐ข2
๐ 2 + ๐ 2๐ฃ2 โค2
3ฮ}. For
any lattice with determinant ฮ, ๐ท1 contains a nonzero lattice point (take the point nearest
the origin and use geometry). ๐ ๐ฅ, ๐ฆ = (๐ ๐ฅ,๐ฆ
๐ ) is an area-preserving map; by applying ๐
to the lattice, we get that every ๐ท๐ has a nonzero lattice point. These points have bounded norm; one norm is hit an infinite number of times. The ratios of these quadratic integers are units.
3 Polynomials 3-1 Polynomials
A polynomial with coefficients in R is a finite combination of nonnegative powers of the variable x. The polynomial ring over R is
๐ ๐ = ๐๐๐ฅ๐
๐
๐=0
|๐๐ โ ๐
with X a formal variable and addition and multiplication defined by: (๐ ๐ฅ = ๐๐๐ฅ๐
๐ , ๐ ๐ฅ = ๐๐๐ฅ
๐๐ )
1. ๐ ๐ฅ + ๐ ๐ฅ = ๐๐ + ๐๐ ๐ฅ๐
๐
2. ๐ ๐ฅ ๐ ๐ฅ = ๐๐๐๐ ๐ฅ๐+๐
๐
R is a subring of R[X] when identified with the constant polynomials in R. The ring of formal
series ๐ [ ๐ ] is defined similarly but combinations need not be finite. Iterating, the ring of polynomials with variables ๐ฅ1, ๐ฅ2, โฆ , ๐ฅ๐ is ๐ [๐ฅ1, ๐ฅ2 , โฆ , ๐ฅ๐ ]. (Formally, use the substitution
principle below to show we can identify ๐ ๐ฅ ๐ฆ with ๐ [๐ฅ, ๐ฆ].) Basic vocab: monomial, degree, constant, leading coefficient, monic Division with Remainder: Let ๐, ๐ โ ๐ [๐] with the leading coefficient of ๐ a unit. There are unique polynomials ๐, ๐ โ ๐ [๐] (the quotient and remainder) so that
๐ ๐ฅ = ๐ ๐ฅ ๐ ๐ฅ + ๐ ๐ฅ , deg ๐ < deg ๐
Substitution Principle: Let ๐: ๐ โ ๐ โฒ be a ring homomorphism. Given ๐1, โฆ , ๐๐ โ ๐ โฒ, there is
a unique homomorphism ฮฆ: ๐ ๐ฅ1, โฆ , ๐ฅ๐ โ ๐ โฒ which agrees with the map ๐ on constant polynomials, and sends ๐ฅ๐ โ ๐๐ . For ๐ = ๐ โฒ, this map is just substituting ๐๐ for the variable ๐ฅ๐ and evaluating the polynomial. A characterization of the GCD: The greatest common divisor ๐ of ๐, ๐ โ ๐ = ๐น[๐ฅ] is the monic polynomial defined by any of the following equivalent conditions:
1. ๐ ๐ = ๐ ๐ + ๐ ๐
2. Of all polynomials dividing f and g, ๐ has the greatest degree. 3. If ๐|๐, ๐ then ๐|๐.
From (1), there exist ๐, ๐ โ ๐ so ๐ = ๐๐ + ๐๐. Adjoining Elements Suppose ๐ผ is an element satisfying no relation other than that implied by ๐ ๐ผ = 0 where ๐ โ ๐ [๐ฅ] and has degree n. Then ๐ ๐ผ โ ๐ ๐ฅ /(๐) is the ring extension. If f is monic, its
distinct elements are the polynomials in ๐ผ (with coefficients in R) of degree less than n, with (1, ๐ผ, โฆ , ๐ผ๐โ1) a basis. A polynomial in ๐ผ is equivalent to its remainder upon division by f. ๐
can be identified with a subring of ๐ [๐ผ] as long as no constant polynomial is identified with 0.
3-2 โ[๐] Tools for factoring in โ[๐]:
1. Inclusion in โ[๐]. 2. Reduction modulo prime p:๐๐ : โ ๐ โ ๐ฝ๐[๐].
A polynomial in โ[๐] is primitive if the greatest common divisor of its coefficients is 1, it is not constant, and the leading coefficient is positive. Gaussโs lemma: The product of primitive polynomials is primitive. Pf. Any prime ๐ in โ is prime in โ[๐ฅ], and a prime divides a polynomial iff it divides every coefficient.
Every nonconstant polynomial ๐ ๐ฅ โ โ[๐ฅ] can be written uniquely as ๐ ๐ฅ = ๐๐0 ๐ฅ , ๐ โ โ and ๐0(๐ฅ) primitive.
If ๐0 is primitive and ๐0|๐ โ โ[๐ฅ] in โ[๐ฅ], then ๐0|๐ in โ[๐ฅ]. If two polynomials have a
common nonconstant factor in โ[๐ฅ] , they have a common nonconstant factor in โ[๐ฅ]. An element of โ[๐ฅ] is irreducible iff it is a prime integer or a primitive irreducible
polynomial in โ[๐ฅ], so every irreducible element of โ[๐ฅ] is prime.
Hence โ[๐ฅ] is a UFD, and each nonzero polynomial can be written uniquely (up to
ordering) as (๐๐ primes, ๐๐ primitive irreducible polynomials) ๐ ๐ฅ = ยฑ๐1 โฏ ๐๐๐1 ๐ฅ โฏ๐๐(๐ฅ)
This technique can be generalized: replace โ with a UFD R, and โ with the field of fractions of R.
By induction, if R is a UFD, then ๐ [๐ฅ1, โฆ , ๐ฅ๐ ] is a UFD.
Rational Roots Theorem: Let ๐ ๐ฅ = ๐๐๐ฅ๐ + โฏ + ๐0 , ๐๐ โ 0. โ๐0
๐1 (in reduced form) is a root
iff ๐1๐ฅ + ๐0 divides f; if ๐1๐ฅ + ๐0 divides f then ๐1|๐๐ and ๐0|๐0.
3-3 Irreducible Polynomials The derivative of a polynomial is formally defined (without calculus) as
๐๐๐ฅ๐
๐
๐=0
โฒ
= ๐๐๐๐ฅ๐โ1
๐
๐=1
.
๐ผ is a multiple root of ๐ ๐ฅ โ ๐น ๐ฅ iff ๐ผ is a root of both ๐(๐ฅ) and ๐ โฒ ๐ฅ . ๐ผ appears as
a root exactly ๐ times if ๐ ๐ ๐ผ = 0 for 0 โค ๐ < ๐ but ๐ ๐ ๐ผ โ 0.
๐, ๐โฒ have a common factor other than 1 iff there is a field extension where f has a multiple root.
If f is irreducible, and ๐ โฒ โ 0, then f has no multiple root (in any field extension). If F has characteristic 0, then f has no multiple root (in any field extension).
If ๐ ๐ฅ = ๐๐๐ฅ๐ + โฏ + ๐0 โ โ[๐ฅ], ๐ โค ๐๐ , and the residue of f modulo p is irreducible in ๐ฝ๐[๐ฅ],
then f is irreducible in โ[๐ฅ]. Irreducible polynomials in ๐ฝ๐[๐ฅ] can be found using the sieve
method.
Eisenstein Criterion: If ๐ ๐ฅ = ๐๐๐ฅ๐ + โฏ + ๐0 โ โ ๐ฅ , ๐ โค ๐๐ , ๐|๐๐โ1, โฆ , ๐0, ๐2 โค ๐0, then f is
irreducible in โ[๐ฅ].
Pf. Factor and take it modulo p; get ๐ = ๐๐๐ฅ๐ (๐๐ ๐ฅ
๐ ). But if ๐, ๐ โ 0, ๐2|๐0.
Extended Eisensteinโs Criterion
Schรถnemannโs Criterion: Suppose
๐ = ๐๐ + ๐๐, ๐ โฅ 1, ๐ prime, ๐, ๐ โ โ[๐ฅ]
deg ๐๐ > deg(๐)
๐ primitive
๐ is irreducible in ๐ฝ๐[๐ฅ]
๐ โค ๐ .
Then k is irreducible in โ[๐ฅ].
Cohnโs Criterion: Let ๐ โฅ 2 and let p be a prime number. Write ๐ = ๐๐๐๐ + โฏ + ๐1๐ + ๐0 in
base b. Then ๐ ๐ = ๐๐๐๐ + โฏ + ๐1๐ + ๐0 is irreducible in โ[๐].
Capelliโs Theorem: Let K be a subfield of โ and ๐, ๐ โ ๐พ[๐]. Let ๐ be a complex root of f and
assume that f is irreducible in ๐พ[๐] and ๐ ๐ โ ๐ is irreducible in ๐พ ๐ [๐]. Then ๐ ๐ ๐ is
irreducible in ๐พ[๐].
[Add proof sketches.]
3-4 Cyclotomic Polynomials A primitive nth root of unity satisfies ๐๐ = 1, but ๐๐ โ 1 for ๐ โ โ, 0 < ๐ < ๐. The nth cyclotomic polynomial is
ฮฆ๐ ๐ = (๐ โ ๐)๐ primitive nth root
.
The cyclotomic polynomial is an irreducible polynomial in โ[๐] of degree ๐(๐). Each
polynomial ๐๐ โ 1 is a product of cyclotomic polynomials:
๐๐ = ฮฆ๐(๐)๐|๐
โ ฮฆ๐ ๐ =๐๐ โ 1
โ ฮฆ๐ ๐ ๐|๐ ,๐<๐
so ฮฆ๐(๐) has integer coefficients. Pf. of irreducibility: Lemma- if ๐ is a primitive nth root of unity and a zero of ๐ โ โ[๐] then
๐๐ is a root for ๐ โค ๐. Proof- Suppose ฮฆ๐ = ๐๐, ๐ ๐ = 0, h irreducible (the minimal polynomial of ๐). ๐๐ is also a zero of ฮฆ๐ . Suppose it is a zero of ๐. Then ๐ is a zero of ๐(๐ฅ๐), so ๐ ๐ฅ๐ = ๐๐. Mod p, ๐ ๐ฅ ๐ = ๐๐, so g and h have a root in cogmmon, contradicting
that ๐๐ โ 1 โ โ๐[๐] has no multiple root (derivative has no common factors with ๐๐ โ 1).
Hence ๐๐ is a zero of h. โ Take powers to different primes to show that all primitive nth roots of unity divide h. Then ๐ = ฮฆ๐ is irreducible.
3-5 Varieties
1. The set ๐ด๐ of n-tuples in a field K is the affine n-space.
2. If S is a set of polynomials in ๐พ ๐1, โฆ , ๐๐ then ๐ ๐ = ๐ฅ โ ๐ด๐ ๐ ๐ฅ = 0 for all ๐ โ ๐
is a (affine) variety.
3. For ๐ โ ๐ด๐ , define the ideal of X to be ๐ผ ๐ = ๐ โ ๐พ ๐1, โฆ , ๐๐ ๐ vanishes on ๐
Note ๐ โ ๐พ ๐1, โฆ , ๐๐ , ๐ โ ๐ด๐ , ๐ ๐ โ ๐ด๐ , ๐ผ ๐ โ ๐พ ๐1, โฆ , ๐๐ . 4. The radical of the ideal I in a ring T is the ideal
๐ผ = {๐ โ ๐ |๐๐ โ ๐ผ for some ๐ โ โ} ๐ด๐ can be made into a topology (the Zariski topology) by taking varieties as closed sets (1-4 below).
Properties: 1. ๐ ๐ = ๐ ๐ผ where I is the ideal generated by S.
2. โฉ ๐ ๐ผ๐ = ๐(โช ๐ผ๐ )
3. If ๐๐ = ๐(๐ผ๐ ) then โ ๐๐๐๐=1 = ๐( ๐1 โฏ ๐๐ ๐๐ โ ๐ผ๐ , 1 โค ๐ โค ๐ ).
4. ๐ด๐ = ๐ 0 , ๐ = ๐(1)
5. ๐ โ ๐ โ ๐ผ ๐ โ ๐ผ(๐), ๐ โ ๐ โ ๐ ๐ โ ๐(๐) 6. ๐ โ ๐ผ๐ ๐ , ๐ โ ๐๐ผ(๐) 7. ๐๐ผ๐ ๐ = ๐ ๐ , ๐ผ๐๐ผ ๐ = ๐ผ(๐)
8. ๐ผ 0 = ๐พ[๐1, โฆ , ๐๐ ] 9. If K is an infinite field, ๐ผ ๐ด๐ = {0}.
a. For n=1, a nonzero polynomial only has finitely many zeros. Extend by induction.
10. ๐ผ ๐1, โฆ , ๐๐ = (๐1 โ ๐1, โฆ , ๐๐ โ ๐๐ ) (the ideal generated by ๐๐ โ ๐๐ ) a. Use division with remainder.
11. If ๐1, โฆ ๐๐ โ ๐ด๐ then ๐ผ = (๐1 โ ๐1, โฆ , ๐๐ โ ๐๐) is a maximal ideal. a. Apply the division algorithm to ๐ โ ๐ฝ\๐ผ.
12. ๐ผ โ ๐ผ๐(๐ผ)
3-6 Nullstellensatz Hilbert Basis Theorem: If R is Noetherian, then ๐ [๐ฅ] is Noetherian. Thus so is ๐ [๐ฅ1, โฆ , ๐ฅ๐ ]. In particular, this holds for ๐ = โ or a field F. Pf. For ๐ผ โ ๐ [๐ฅ] an ideal, the set whose elements are the leading coefficients of the polynomials in I (including 0) forms the ideal of leading coefficients in R. Take generators for this ideal and polynomials with those leading coefficients, multiplying by x as necessary so they have the same degree n. The polynomials with degree less than n form a free, Noetherian R-module P with basis (1, ๐ฅ, โฆ ๐ฅ๐โ1). ๐ โฉ ๐ผ has a finite generating set. Put the two generating sets together. The ring of formal power series ๐ [ ๐ ] is also Noetherian. Noether Normalization Lemma: Let A be a finitely generated K-algebra. There exists a
subset ๐ฆ1, โฆ , ๐ฆ๐ of A such that the ๐ฆ๐ are algebraically independent over K and A is integral over ๐พ[๐ฆ1, โฆ , ๐ฆ๐] (all elements of A are algebraic integers over ๐พ[๐ฆ1, โฆ , ๐ฆ๐]). Pf. Induct on n; n=1 trivial. Take a maximally algebraically independent subset ๐ฅ1, โฆ , ๐ฅ๐ โ{๐ฅ1, โฆ , ๐ฅ๐}; assume ๐ > ๐. By algebraic dependency, there exists ๐ โ ๐พ ๐1, โฆ , ๐๐ so that ๐ ๐ฅ1, โฆ , ๐ฅ๐ = 0. Lexicographically order the monomials, and choose weights ๐ค๐ to match
the lexicographic order. Set ๐ฅ๐ = ๐ง๐ + ๐ฅ๐๐ค๐ . Then we get a polynomial in ๐ฅ๐ , where term with
the highest power of ๐ฅ๐ is uncancelled. ๐ฅ๐ is integral over ๐พ[๐ง1, โฆ , ๐ง๐โ1]; finish by induction.
Cor. If B is a finitely generated K-algebra, and I is a maximal ideal, then ๐ต ๐ผ is a finite
extension of K (K is embedded via ๐ โ ๐ + ๐ผ.): In the above, we must have ๐ = 0; B/I is algebraic over K.
Hilbertโs Nullstellensatz: For any field K and ๐ โ โ, the following are equivalent: 1. K is algebraically closed. 2. (Maximal Ideal Theorem) The maximal ideals of ๐พ[๐1, โฆ , ๐๐ ] are the ideals of the
form (๐1 โ ๐1, โฆ , ๐๐ โ ๐๐). 3. (Weak Nullstellensatz) If I is a proper ideal of ๐พ[๐1, โฆ , ๐๐], then ๐(๐ผ) is not empty.
4. (Strong Nullstellensatz) If I is an ideal of ๐พ[๐1, โฆ , ๐๐ ] then ๐ผ๐ ๐ผ = ๐ผ. (2)โ(3): For I a proper ideal, let J be a maximal ideal containing it. Then ๐ ๐ฝ โ ๐(๐ผ) so ๐ฝ = (๐1 โ ๐1 , โฆ , ๐๐ โ ๐๐), ๐ โ ๐(๐ฝ).
(3)โ(4): Rabinowitsch Trick: 1. Let ๐ โ ๐ผ๐ ๐ผ . Let ๐1, โฆ , ๐๐ be a generating set for ๐พ[๐1, โฆ , ๐๐ , ๐]. Let ๐ผโ be the ideal
generated by ๐1, โฆ , ๐๐ , 1 โ ๐๐.
2. ๐ ๐ผโ = ๐ a. If ๐1, โฆ , ๐๐ , ๐๐+1 โ ๐ด๐+1 and ๐1, โฆ , ๐๐ โ ๐ ๐ผ , then ๐1, โฆ , ๐๐ , ๐๐+1 โ ๐(๐ผโ)
since it is not a zero of 1 โ ๐๐. b. If ๐1, โฆ , ๐๐ โ ๐ ๐ผ then ๐1, โฆ , ๐๐ , ๐๐+1 โ ๐(๐ผโ) (an even weaker statement).
3. Using the weak Nullstellensatz (see below), we can write
1 = ๐๐๐๐ + ๐ 1 โ ๐๐
๐
๐=1
since ๐ ๐ผโ = ๐ โ 1 โ ๐ผโ = ๐พ[๐1, โฆ , ๐๐ , ๐]. 4. Set ๐ = 1/๐, and multiply to clear denominators
๐๐ = ๐๐ ๐1, โฆ , ๐๐ ๐๐ ๐1, โฆ ๐๐
๐
๐=1
โ ๐ผ
(4)โ(3): The radical of an ideal is the intersection of all prime ideals containing I. I is in a
maximal, prime ideal P; so is ๐ผ, so ๐ผ is proper. ๐ผ๐(๐ผ) is proper by (4), so ๐ ๐ผ โ ๐.
(3)โ(2): For I maximal, there is ๐ โ ๐(๐ผ). Since I is maximal, it must be in (๐1 โ ๐1, โฆ , ๐ฅ๐ โ๐๐).
(1)โ(2): Let I be a maximal ideal. K can be imbedded via ๐ โ ๐ + ๐ผ in ๐พ ๐1, โฆ , ๐๐ ๐ผ ; by the corollary to Noether Normalization Lemma, this is a finite extension of K so must be K (since
it is algebraically closed). Then ๐๐ + ๐ผ = ๐๐ + ๐ผ โ ๐๐ โ ๐๐ โ ๐ผ, ๐1 โ ๐1 , โฆ , ๐๐ โ ๐๐ โ ๐ผ, with equality since the LHS is maximal. (2)โ(1): If f is a nonconstant polynomial in ๐พ[๐1] with no root in K, regard it as a polynomial
in ๐พ[๐1, โฆ , ๐๐ ] with no root in ๐ด๐ . Then I is in a maximal ideal (๐1 โ ๐1, โฆ , ๐๐ โ ๐๐), so has
root ๐1 = ๐1. Combinatorial Nullstellensatz: Let F be a field, ๐ โ ๐น[๐1, โฆ , ๐๐ ] and let ๐1, โฆ , ๐๐ be nonempty subsets of F.
1. If ๐ ๐ 1, โฆ , ๐ ๐ = 0 for all ๐ 1, โฆ , ๐ ๐ โ ๐1 ร โฏ ร ๐๐ (i.e. ๐ โ ๐(๐1 ร โฏ ร ๐๐)) then ๐ lies in the ideal generated by the polynomials ๐๐ ๐๐ = โ (๐๐ โ ๐ )๐ โ๐๐
. The polynomials
๐1, โฆ , ๐๐ satisfying ๐ = ๐1๐1 + โฏ + ๐๐๐๐
can be chosen so that deg ๐๐ โค deg ๐ โ deg(๐๐) for all i. If ๐1, โฆ , ๐๐ โ ๐ [๐1, โฆ , ๐๐] for some subring ๐ โ ๐น, we can choose ๐๐ โ ๐ [๐1, โฆ , ๐๐].
a. The ๐ โ ๐๐ are all zeros of ๐๐ of degree |๐๐| so by subtracting multiples of ๐๐
every ๐๐๐ in f can be replaced with a linear combination of 1, โฆ , ๐ฅ๐
๐๐ โ1. Then by
counting zeros the result is actually 0; i.e. f is in the form above. 2. If deg ๐ = ๐ก1 + โฏ + ๐ก๐ where ๐ก๐ are nonnegative integers with ๐ก๐ < |๐๐|, and if the
coefficient of ๐1๐ก1 โฏ ๐๐
๐ก๐ is not 0, then there exist ๐ ๐ โ ๐๐ so that ๐ ๐ 1, โฆ , ๐ ๐ โ 0, i.e.
๐ โ ๐ โ ๐(๐1 ร โฏ ร ๐๐).
4 Fields Examples of Fields An extension K of a field F (also denoted ๐พ/๐น) is a field containing F.
Number field: subfield of โ
Finite field: finitely many elements
Function field: Extensions of โ(๐ก) of rational functions
4-1 Fundamental Theorem of Algebra Fundamental Theorem of Algebra: Every nonconstant polynomial with complex coefficients has a complex zero. Thus, the field of complex numbers is algebraically closed, and every polynomial in โ[๐ฅ] splits completely.
Pf. Let ๐ ๐ฅ = ๐ฅ๐ + ๐๐โ1๐ฅ๐โ1 + โฏ + ๐1๐ฅ + ๐0. Let ๐ฅ = ๐๐๐๐ - the parameterization of a circle about the origin. ๐ ๐ฅ โ ๐ฅ๐ is small for large r, so for large r, ๐(๐ฅ) winds around the origin n
times as ๐ goes from 0 to 2๐๐.2 (f is a person walking a dog on a circular path n times around the origin. The dog will also walk around the origin n times provided the leash is shorter than the radius.) For small r, ๐ ๐ฅ makes a small loop around ๐0, and winds around
the origin 0 times. For some intermediate value, ๐ ๐ฅ will pass through the origin.
4-2 Algebraic Elements
Let K be an extension, and ๐ผ be an element of K. ๐ผ is algebraic over F if it is the zero of a polynomial in ๐น[๐ฅ], and transcendental otherwise. ๐ผ is transcendental iff the substitution
homomorphism ๐: ๐น ๐ฅ โ ๐พ is injective. ๐น ๐ผ is the smallest subfield of K that contains F and ๐ผ. The irreducible polynomial ๐ of ๐ผ
over F is the monic polynomial of lowest degree in ๐น[๐ฅ] having ๐ผ as a zero.
Any polynomial in ๐น[๐ฅ] having ๐ผ as a zero divides ๐.
๐น ๐ฅ /(๐) is an extension field of F with ๐ฅ a root of ๐ ๐ฅ = 0. The substitution map ๐น ๐ฅ /(๐) โ ๐น[๐ผ] is an isomorphism, so ๐น ๐ฅ /(๐) โ ๐น(๐ผ). [*]
For every polynomial in F[x], there is an extension field in which it splits (factors) completely, i.e. every polynomial has a splitting field.
If f has degree ๐ then ๐น(๐ผ) is a vector space with dimension ๐ and basis
(1, ๐ผ, โฆ , ๐ผ๐โ1).
Let ๐ผ, ๐ฝ be elements of ๐พ/๐น, ๐ฟ/๐น algebraic over F. There is an isomorphism of fields ๐น ๐ผ โ ๐น(๐ฝ) sending ๐ผ โ ๐ฝ iff ๐ผ, ๐ฝ have the same irreducible polynomial.
Let ๐พ, ๐ฟ be extensions of F. A F-isomorphism is an isomorphism ๐: ๐พ โ ๐ฟ that restricts to the identity on F. Then K and L are isomorphic as field extensions. If ๐ผ is a zero of ๐ in ๐พ, then ๐(๐ผ) is a zero of ๐.
4-3 Degree of a Field Extension The degree [๐พ: ๐น] is the dimension of K as an F-vector space.
If ๐ผ is algebraic over F, then [๐น ๐ผ : ๐น] is the degree of the irreducible polynomial of ๐ผ
over F, and if ๐ผ is transcendental, ๐น ๐ผ : ๐น = โ.
๐พ: ๐น = 1 iff ๐น = ๐พ.
2 i.e. it is homotopic to the loop going around the origin n times in โ โ 0 .
If ๐พ: ๐น = 2 then K can be obtained by adjoining a square root ๐ฟ of an element in F: ๐ฟ2 โ ๐น
Multiplicative property: If ๐น โ ๐พ โ ๐ฟ, then ๐ฟ: ๐น = ๐ฟ: ๐พ [๐พ: ๐น]. o If K is a finite extension of F, and ๐ผ โ ๐พ, then the degree of ๐ผ divides [๐พ: ๐น]. o If ๐ผ โ ๐ฟ is algebraic over F, it is algebraic over ๐พ with degree less than or
equal to its degree over ๐น. o A field extension generated by finitely many algebraic elements is a finite
extension. o The set of elements of K that are algebraic over F is a subfield of K.
Let L be an extension field of F, and let ๐พ, ๐นโฒ be subfields of L that are finite
extensions of F. Let ๐พโฒ : ๐น = ๐, ๐พ: ๐น = ๐, ๐นโฒ : ๐น = ๐. Then m and n divide N, and ๐ โค ๐๐.
Finding the Irreducible Polynomial of ๐พ (The dum way) Compute powers of ๐พ, and find a relation between them. (The smart way)
1. If ๐พ = ๐1 + โฏ + ๐๐ , ๐1 โฏ ๐๐ where each ๐1 is algebraic (for example a nth root), then
its conjugates (other zeros of the irreducible polynomial) are in the form ๐1 + โฏ +๐๐ , ๐1 โฏ ๐๐ , respectively where ๐๐ is a conjugate of ๐๐ . (Not all these may be conjugates.)
2. The irreducible polynomial is โ (๐ฅ โ ๐พโฒ)๐พ โฒ conjugate of ๐พ . [Note: This gives an elementary
proof of the fact that the algebraic numbers/ integers form a field/ ring. For ๐, ๐ algebraic, expand the product (๐ฅ โ ๐โฒ โ ๐โฒ ), ๐โฒ , ๐โฒ running over the conjugates of ๐, ๐, and use the Fundamental Theorem on Symmetric Polynomials.]
4-4 Application: Constructions! Rules:
1. Two points (0,0) and (1,0) are given (constructed). 2. If P, Q have been constructed, we can draw (construct)
a. the line through them b. a circle with center at P and passing through Q.
3. Points of intersection of constructed lines and circles are constructed. 4. A number is constructible if (a,0) is constructible.
Finding all constructible numbers: 1. If ๐ = ๐0, ๐1 , ๐ = (๐0 , ๐1) have coordinates in F, then the line in (2a) is defined by a
linear equation with coefficients in F, while the circle in (2b) is defined by a quadratic equation with coefficients in F.
2. The point of intersection of two lines/ circles whose equations have coefficients in F has coordinates in a real quadratic extension of F.
3. Let P be a constructible point. There is a chain of fields โ = ๐น0 โ ๐น1 โ โฏ โ ๐น๐ = ๐พ such that
a. K is a subfield of โ. b. The coordinates of P are in ๐น๐ . c. ๐น๐+1: ๐น๐ = 2, and ๐พ: โ = 2๐ .
๐พโฒ
๐น
๐นโฒ ๐พ ๐ โค ๐๐
๐ ๐
โค ๐ โค ๐
Thus a constructible number has degree over โ a power of 2. 4. Conversely, if a chain of fields satisfies the conditions above, then every element of K
is constructible. Examples:
1. Trisecting the angle with compass and straightedge is impossible. cos 20ยฐ is not constructible.
2. For p prime, a regular p-gon can be constructed iff ๐ = 2๐ + 1.
4-5 Finite Fields A finite field is a vector space over ๐ฝ๐ for some prime p, so has order ๐ = ๐๐ . The (unique)
field of order q is denoted by ๐ฝ๐ .
1. The elements in a field of order q are roots of ๐ฅ๐ โ ๐ฅ = 0 (everything is modulo p). a. The multiplicative group ๐ฝ๐
ร of nonzero elements has order ๐ โ 1. The order of
any element divides ๐ โ 1 so ๐ผ๐โ1 = 0 for any ๐ผ โ ๐ฝ๐.
2. ๐ฝ๐ร is a cyclic group of order ๐ โ 1.
a. By the Structure Theorem for Abelian Groups, ๐ฝ๐ร is a direct product of cyclic
subgroups of orders ๐1| โฏ |๐๐ , and the group has exponent ๐๐ . ๐ฅ๐๐ โ 1 = 0
has at most ๐๐ roots, so ๐ = 1, ๐๐ = ๐ โ 1. 3. There exists a unique field of order q (up to isomorphism).
a. Existence: Take a field extension where ๐ฅ๐ โ ๐ฅ splits completely. If ๐ผ, ๐ฝ are roots of ๐ฅ๐ โ ๐ฅ = 0 then ๐ผ + ๐ฝ ๐ = ๐ผ + ๐ฝ. Since โ1 is a root, โ๐ผ is a root. The roots form a field.
b. Uniqueness: Suppose ๐พ, ๐พโฒ have order ๐. Let ๐ผ be a generator of ๐พร; ๐พ = ๐น(๐ผ). The irreducible polynomial ๐ โ ๐พ[๐ฅ] with root ๐ผ divides ๐ฅ๐ โ ๐ฅ.
๐ฅ๐ โ ๐ฅ splits completely in both ๐พ, ๐พโฒ, so f has a root ๐ผโฒ โ ๐พโฒ. Then ๐น ๐ผ โ ๐น ๐ฅ /(๐) โ ๐น(๐ผโฒ) = ๐พโฒ.
4. A field of order ๐๐ contains a subfield of order ๐๐ iff ๐|๐. (Note this is a relation between the exponents, not the orders.)
a. ๐ฝ๐๐ โ ๐ฝ๐๐ โ ๐|๐: Multiplicative property of the degree.
b. ๐ฝ๐๐ โ ๐ฝ๐๐ โ ๐|๐: ๐๐ โ 1|๐๐ โ 1. Cyclic ๐ฝ๐๐ร contains a cyclic group of order ๐๐ .
Including 0, they are the roots of ๐ฅ๐๐โ ๐ฅ = 0 and thus form a field by 3a.
5. The irreducible factors of ๐ฅ๐ โ ๐ฅ over ๐ฝ๐ are the irreducible polynomials ๐ in ๐น[๐ฅ]
whose degrees divide r.
a. โ: Multiplicative property b. โ: Let ๐ฝ be a root of g. If ๐|๐, by (4), ๐ฝ๐ contains a subfield isomorphic to
๐น(๐ฝ). ๐ has a root in ๐ฝ๐ so divides ๐ฅ๐ โ ๐ฅ.
6. For every r there is an irreducible polynomial of degree r over ๐ฝ๐ .
a. ๐ฝ๐(๐ = ๐๐) has degree r over ๐ฝ๐ , and has a cyclic multiplicative group
generated by an element ๐ผ. ๐ฝ๐(๐ผ) has degree r over ๐ฝ๐ .
To compute in ๐ฝ๐ , take a root ๐ฝ of the irreducible factor of ๐ฅ๐ โ ๐ฅ of degree r; (1, ๐ฝ, โฆ , ๐ฝ๐โ1)
is a basis. Let ๐๐(๐) be the number of irreducible monic polynomials of degree d in ๐ฝ๐ . Then by (2),
๐๐ = ๐๐๐(๐)๐|๐
.
By Mรถbius inversion,
๐๐ ๐ =1
๐ ๐
๐
๐ ๐๐
๐|๐
.
Primitive elements Let K be a field extension of F. An element ๐ผ โ ๐พ such that ๐พ = ๐น(๐ผ) is a primitive element for the extension. Primitive Element Theorem: Every finite extension of a field F contains a primitive element. Proof when F has characteristic 0: Show that if ๐พ = ๐น(๐ผ, ๐ฝ) with ๐ผ, ๐ฝ โ ๐พ then ๐พ = ๐ฝ + ๐๐ผ is a primitive element for K over F. Let ๐, ๐ be the irreducible polynomials of ๐ผ, ๐ฝ, and ๐ผ๐ , ๐๐ be
the conjugates of ๐ผ, ๐ฝ. For all but finitely many ๐, the numbers ๐ฝ๐ + ๐๐ผ๐ are all distinct. For
such a value of ๐, to show that ๐พ = ๐ฝ + ๐๐ผ is a primitive element, consider ๐ ๐ฅ =๐ ๐พ โ ๐๐ฅ โ ๐น(๐พ). gcd ๐, ๐ = ๐ฅ โ ๐ผ by choice of c, so ๐ผ โ ๐น(๐พ) as desired.
5 Modules
More structureโ
No division Division
More complex โ
Ring Field
Module Vector Space
Algebra Division Algebra
5-1 Modules
A left/right R-module ๐๐ /๐๐ over the ring R is an abelian group (M,+) with addition and
scalar multiplication (๐ ร ๐ โ ๐ or ๐ ร ๐ โ ๐) defined so that for all ๐, ๐ โ ๐ and ๐ฅ, ๐ฆ โ ๐,
Left Right
1. Distributive ๐ ๐ฅ + ๐ฆ = ๐๐ฅ + ๐๐ฆ ๐ฅ + ๐ฆ ๐ = ๐ฅ๐ + ๐ฆ๐
2. Distributive ๐ + ๐ ๐ฅ = ๐๐ฅ + ๐ ๐ฅ ๐ฅ ๐ + ๐ = ๐ฅ๐ + ๐ฅ๐
3. Associative ๐ ๐ ๐ฅ = ๐๐ ๐ฅ ๐ฅ๐ ๐ = ๐ฅ(๐๐ )
4. Identity 1๐ฅ = ๐ฅ ๐ฅ1 = ๐ฅ
A (S,R)-bimodule ๐๐ ๐ has both the structure of a left S-module and right R-modules.
Modules are generalizations of vector spaces. All results for vector spaces hold except ones
depending on division (existence of inverse in R). ๐ โ ๐ is linearly dependent if there exist
๐ฃ1, โฆ , ๐ฃ๐ โ ๐ such that ๐๐๐ฃ๐ โ 0 and ๐1๐ฃ1 + โฏ + ๐๐๐ฃ๐ = 0.
Again, a basis is a linearly independent set that generates the module. Note that if elements are linearly independent, it is not necessary that one element is a linear combination of the others, and bases do not always exist. Every basis for V (if it exists) contains the same number of elements. V is finitely generated if it contains a finite subset spanning V. The rank is the size of the smallest generating set. A submodule of a R-module is a nonempty subset closed under addition and scalar
multiplication. Viewing R as an R-module, the submodules of R are the ideals in R. In โ๐ , submodules correspond to lattices, with the area/volume of a fundamental parallelogram/parallelepiped equal to the determinant. A free module with n generators has a basis with n elements. It is isomorphic to ๐ ๐ . An isomorphism preserves addition and scalar multiplication. Unlike vector spaces, not all
finitely generated modules are isomorphic to some ๐ ๐ . Basic Theorems:
1. If W is a submodule of V, the quotient module ๐/๐ is a R-module, and the canonical
map ๐: ๐ โ ๐/๐ is a homomorphism. 2. Mapping Property: Let ๐: ๐ โ ๐โฒ be a homomorphism of R-modules with kernel
containing W. There is a unique homomorphism ๐ with ๐ = ๐ โ ๐.
3. First Isomorphism Theorem: If ๐ is surjective with kernel W, ๐ is an isomorphism.
4. Correspondence Theorem: Let ๐: ๐ โ ๐โฒ be a surjective homomorphism. There is a bijective correspondence between submodules of V' and submodules of V that containing ker ๐ = ๐: ๐ with ๐ โ ๐ โ ๐ is associated with ๐(๐); ๐/๐ โ ๐ โฒ/๐(๐).
V'
V G
๐
๐ ๐
5. Second Isomorphism Theorem: Let ๐ and ๐ be submodules of ๐, and let ๐ + ๐ = ๐ฅ + ๐ฆ โถ ๐ฅ โ ๐, ๐ฆ โ ๐ . Then ๐ + ๐ and ๐ โฉ ๐ are submodules of ๐ and
๐ + ๐
๐โ
๐
๐ โฉ ๐
6. Third Isomorphism Theorem: Let ๐ โ ๐ฟ โ ๐ be modules. Then ๐/๐ฟ โ (๐/๐)/(๐ฟ/๐).
5-2 Structure Theorem
Matrices, invertible matrices, the general linear group, the determinant, and change of bases matrices all generalize to a ring R. However, a R-matrix A is invertible iff its determinant is a unit.
Properties of matrices in a field such as det(๐ด๐ต) = det ๐ด det ๐ต continue to hold in a ring, because they are polynomial identities. Smith Normal Form For a matrix over an Euclidean domain R [*] (such as โ or ๐น[๐ก]), elementary row/ column operations correspond to left and right multiplication by elementary matrices and include:
(1) Interchanging 2 rows/ columns (2) Multiplying any row/ column by a unit (3) Adding any multiple of a row/ column to another row/ column
However, note arbitrary division in R is illegal. A ๐ ร ๐ matrix is in Smith (or Hermite) normal form if
1. It is diagonal. 2. The entries ๐1, โฆ , ๐๐ on the main diagonal satisfy ๐๐|๐๐+1, 1 โค ๐ < min(๐, ๐). (Ones
at the end may be 0.)
Every matrix is equivalent to a unique matrix N in normal form. For a ๐ ร ๐ matrix A, follow this algorithm to find it:
1. Make the first column
๐0โฎ0
.
a. Choose the nonzero entry ๐ in the first column that has the least norm.
b. For each other nonzero entry ๐, use division to write ๐ = ๐๐ + ๐, where ๐ is the remainder upon division. Subtract ๐ times the row with ๐ from the row with ๐.
c. Repeat a and b until there is (at most) one nonzero entry. Switch the first row with that row if necessary.
2. Put the first row in the form ๐ 0 โฏ 0 by following the steps above but exchanging the words โrowsโ and โcolumnsโ.
3. Repeat 1 and 2 until the first entry ๐ is the only nonzero entry in its row and column. (This process terminates because the least degree decreases at each step.)
4. If ๐ does not divide every entry of A, find the first column with an entry not divisible by g and add it to column 1, and repeat 1-4; the degree of โgโ will decrease. Else, go to the next step.
5. Repeat with the ๐ โ 1 ร (๐ โ 1) matrix obtained by removing the first row and column.
Solving ๐ด๐ = ๐ต in R: 1. Write ๐ด = ๐๐ดโฒ๐โ1, where ๐ดโฒ is in normal form. Suppose the nonzero diagonal entries
are ๐1, ๐2, โฆ , ๐๐ . 2. The solutions ๐โฒ of the homogeneous system ๐ดโฒ๐โฒ = 0 are the vectors whose first k
coordinates are 0.
3. The solutions of ๐ด๐ = 0 are in the form ๐๐ = ๐๐โฒ.
4. The equation has a solution iff ๐ต is in the form ๐๐โฒ , ๐โฒ =
๐1๐1
โฎ๐๐๐๐
0โฎ
. Use linear algebra to
find a particular solution ๐๐ . (The condition guarantees that the entries are in R.)
Then the solutions are ๐๐ + ๐๐ , for ๐๐ a homogeneous solution.
Structure Theorem: (a.k.a. Fundamental Decomposition Theorem)
Let M be a finitely generated module over the PID R (such as โ or ๐น[๐ก]). Then M is a direct sum of cyclic modules and a free module ๐ถ1 โ โฏ โ ๐ถ๐ โ ๐ฟ, where ๐ถ๐ โ ๐ /(๐๐). The cyclic modules can be chosen to satisfy either of the following conditions:
1. ๐1|๐2 โฏ |๐๐ 2. Each ๐๐ is the power of an irreducible element.
5-3 Noetherian and Artinian Rings The following conditions on a R-module V are equivalent:
1. Every submodule is finitely generated.
2. Ascending chain condition: There is no infinite strictly increasing chain ๐1 โ ๐2 โ โฏ of submodules. (Pf. of โ: Union of chain finitely generated.)
A ring is Noetherian if every ideal of R is finitely generated. In a Noetherian ring, every proper ideal is contained in a maximal ideal.
Let ๐ be a homomorphism of R-modules. 1. If V is finitely generated and ๐ is surjective, then ๐โฒ is finitely generated. 2. If ker ๐ , im ๐ are finitely generated, so is V. (Pf. Take a generating set for the kernel
and some preimage of a generating set for the image.)
3. In particular, if ๐ is finitely generated, so is ๐/๐. If ๐/๐ and ๐ are finitely generated, so is ๐.
If R is Noetherian, then every submodule of a finitely generated R-module is finitely generated.
Pf. Using a surjective map ๐: ๐ ๐ โ ๐, it suffices to prove it for ๐ ๐ . Induct on m. For the
projection ๐: ๐ ๐ โ ๐ ๐โ1, the image and kernel are finitely generated. A ring is Artinian if it satisfies the descending chain condition on ideals: There is no infinite
strictly decreasing chain ๐ผ1 โ ๐ผ2 โ โฏ of ideals.
5-4 Application 1: Abelian Groups
An abelian group corresponds to a โ-module with integer multiplication defined by ๐๐ฃ = ๐ฃ + ๐ฃ + โฏ + ๐ฃ
๐ times
. Abelian groups and โ-modules are equivalent concepts, so
generalizing linear algebra for modules helps us study abelian groups. If W is a free abelian group of rank m, and U is a subgroup, then U is a free abelian group of
rank at most ๐. Pf. Choose a (finite) set of generators ๐ฝ = (๐ข1, โฆ , ๐ข๐) for U and a basis ๐พ = (๐ค1, โฆ , ๐ค๐ ) for
W. Write ๐ข๐ = ๐ค๐๐๐๐๐ . Then left multiplication by A is an surjective homomorphism โ๐ โ
โ๐ . Diagonalizing A, we can find an explicit basis for U with at most ๐ โค ๐ elements.
Generators and Relations
Let A be a matrix, and let ๐ด๐ ๐ denote the image of ๐ ๐ under left multiplication by ๐ด. The module ๐ = ๐ ๐ /๐ด๐ ๐ is presented by the matrix A.
An abelian group G with n generators ๐ฃ1, โฆ , ๐ฃ๐ can be identified with a quotient module of
โ๐ . The set of ๐1, โฆ , ๐๐ ๐ โ โ๐ such that
๐๐๐ฃ๐
๐
๐=1
= 0
forms a submodule of โ๐ . They are the relations in G, and form the kernel of the map
โ๐ โ ๐บ. If ๐๐1 , โฆ , ๐๐๐ ๐ โ โ๐ generate this submodule, letting A be the matrix with these rows,
๐๐๐ฃ๐
๐
๐=1
= 0 โ ๐1
โฎ๐๐
= ๐ด๐ฃ for some ๐ฃ โ โ๐
Then G corresponds to the module โ๐/๐ดโ๐ ; G is presented by A. An abelian group that is presented by a matrix needs only satisfy the relations implied (through linearity) by the relations given by the columns of A. To determine a group from its presentation:
1. Use elementary row and column operations to write A in normal form. 2. Delete any columns of 0โs (trivial relations). 3. If 1 is the only nonzero entry in its column, delete that row and column. (This is a
relation of the form ๐ฃ = 0.)
4. If the diagonal entries are ๐1, โฆ , ๐๐ , and there are ๐ zero rows, then the group is
isomorphic to โ๐1โ โฏ โ โ๐๐
โ โ๐ โ ๐ถ๐1โ โฏ โ ๐ถ๐๐
โ ๐๐ถโ .
Structure Theorem for Finitely Generated Abelian Groups: a.k.a Unicity Theorem for Abelian Group Decomposition, Basis Theorem
A finitely generated abelian group V is the direct sum of cyclic subgroups and a free abelian group: ๐ = ๐ถ๐1
โ โฏ โ ๐ถ๐๐โ ๐ฟ. The orders can be chosen uniquely to satisfy either of the
following two conditions:
1. 1 < ๐1|๐2 โฏ |๐๐ 2. All the ๐๐ are prime power orders. (Uniqueness follows from counting orders in p-
groups.)
5-5 Application 2: Linear Operators A linear operator T on a F-vector space corresponds to a ๐น[๐ก]-module with multiplication by a polynomial defined by ๐ก๐ฃ = ๐ ๐ฃ , ๐ ๐ก ๐ฃ = ๐ ๐ (๐ฃ). A submodule corresponds to a T-invariant subspace. The structure theorem gives: Rational Canonical Form: Every linear operator T on finite-dimensional V has a rational canonical form.
๐ ๐ฝ =
๐ถ1 ๐๐ ๐ถ2
โฏ ๐โฏ ๐
โฎ โฎ๐ ๐
โฎโฏ ๐ถ๐
where each ๐ถ๐ is the companion matrix of an invariant factor ๐๐ . The rational canonical form
โ๐
UL
W
๐ถ
๐
๐ต
โ๐ ๐ด
is unique under the condition ๐๐+1|๐๐ for each 1 โค ๐ < ๐.
The companion matrix of the monic polynomial ๐ ๐ก = ๐0 + ๐1๐ก + โฏ + ๐๐โ1๐ก๐โ1 + ๐ก๐ is
๐ถ(๐) =
0 01 00 1
โฏ 0 โ๐0
โฏ 0 โ๐1
โฏ 0 โ๐2
โฎ โฎ0 0
โฑ โฎ โฎโฏ 1 โ๐๐โ1
because the characteristic polynomial of C(p) is โ1 ๐๐(๐ก).
The product of the invariant factors is the characteristic polynomial of T.
5-6 Polynomial Rings in Several Variables
Let ๐ = โ ๐ฅ1 , โฆ , ๐ฅ๐ = โ ๐ ๐ โ โ๐ , and V be a finitely generated R-module with presentation matrix ๐ด(๐). V is a free module of rank ๐ iff ๐ด(๐) has rank ๐ โ ๐ at every point
๐ โ โ๐ . The subspace ๐(๐) spanned by the columns varies continuously as c if the dimension does not โjump around.โ Continuous families of vector spaces are vector bundles. V is free iff
๐(๐) forms a vector bundle over โ๐ .
5-7 Tensor Products Commutative Rings: The tensor product ๐ โ๐ ๐ of R-modules M and N is the (unique) R-module T (along with a bilinear map ๐: ๐ ร ๐ โ ๐) satisfying the Universal Mapping Property: for any bilinear map
๐: ๐ ร ๐ โ ๐, there is a unique R-homomorphism ๐: ๐ โ ๐ such that ๐ = ๐๐.
One way to construct it is as follows: Let F be the free module with basis ๐ ร ๐, and let ๐บ be the submodule generated by ๐ฅ + ๐ฅโฒ , ๐ฆ โ ๐ฅ, ๐ฆ โ (๐ฅโฒ , ๐ฆ), ๐ฅ, ๐ฆ + ๐ฆ โฒ โ ๐ฅ, ๐ฆ โ (๐ฅ, ๐ฆ โฒ), ๐๐ฅ, ๐ฆ โ ๐(๐ฅ, ๐ฆ), ๐ฅ, ๐๐ฆ โ ๐(๐ฅ, ๐ฆ) for all ๐ฅ, ๐ฅโฒ โ ๐; ๐ฆ, ๐ฆ โฒ โ ๐, ๐ โ ๐ . Then ๐ โ๐ ๐ = ๐น/๐บ;
๐ฅ โ ๐ฆ denotes ๐ฅ, ๐ฆ + ๐บ. The relations make ๐ฅ โ ๐ฆ linear: 1. ๐ฅ + ๐ฅโฒ โ ๐ฆ = ๐ฅ โ ๐ฆ + ๐ฅโฒ โ ๐ฆ
2. ๐ฅ โ ๐ฆ + ๐ฆ โฒ = ๐ฅ โ ๐ฆ + ๐ฅ โ ๐ฆ โฒ 3. ๐ ๐ฅ โ ๐ฆ = ๐๐ฅ โ ๐ฆ = ๐ฅ โ ๐๐ฆ
Note that in general, an element of ๐ โ ๐ is a sum (linear combination) of elements in the
form ๐ฅ โ ๐ฆ. Basic properties (prove using UMP)
1. ๐ โ ๐ โ ๐ โ ๐ 2. ๐ โ ๐ โ ๐ โ ๐ โ (๐ โ ๐)
3. ๐ โ ๐ โ ๐ โ ๐ โ ๐ โ (๐ โ ๐)
4. ๐ โ๐ ๐ โ ๐, ๐ ๐ โ ๐ โ ๐๐ where ๐ด๐ means the direct sum of m copies of A. 5. ๐ ๐ โ ๐ ๐ = ๐ ๐๐ .
The tensor product ๐1 โ ๐2 of homomorphisms ๐1: ๐1 โ ๐1, ๐2: ๐2 โ ๐2 is the map ๐: ๐1 โ๐2 โ ๐1 โ ๐2 such that ๐ ๐ฅ1 โ ๐ฅ2 = ๐1 ๐ฅ1 โ ๐2(๐ฅ2). Note that ๐1 โ ๐2 โ ๐1 โ ๐2 = ๐1 โ ๐1 โ (๐2 โ ๐2). When applied to the linear transformations corresponding to the
homomorphisms, the tensor (Kronecker) product of ๐ ร ๐ matrix A and ๐ ร ๐ matrix B is
๐ ร ๐ ๐
๐ ๐
๐
๐
๐ดโจ๐ต =
๐11๐ต โฏ ๐1๐๐ต
โฎ โฑ โฎ๐๐1๐ต โฏ ๐๐๐ ๐ต
. The ordering of the basis is ๐ฃ1 โ ๐ค1, โฆ , ๐ฃ1 โ ๐ค๐ , โฆ , ๐ฃ๐ โ ๐ค๐ .
For the tensor product of algebras, multiplication is given by ๐ โ ๐ ๐โฒ โ ๐โฒ = ๐๐โฒ โ ๐๐โฒ. Noncommutative rings: The tensor product ๐ โ๐ ๐ of right R-module ๐๐ and left R-module ๐๐
is an abelian
group T (along with a bilinear map ๐: ๐ ร ๐ โ ๐) satisfying the Universal Mapping Property: for any biadditive, R-balanced map ๐: ๐ ร ๐ โ ๐
1. ๐ ๐ฅ + ๐ฅโฒ , ๐ฆ = ๐ ๐ฅ, ๐ฆ + ๐ ๐ฅโฒ , ๐ฆ , ๐ ๐ฅ, ๐ฆ + ๐ฆ โฒ = ๐ ๐ฅ, ๐ฆ + ๐ ๐ฅ, ๐ฆ โฒ 2. ๐ ๐ฅ๐, ๐ฆ = ๐(๐ฅ, ๐๐ฆ)
there is a unique abelian group homomorphism ๐: ๐ โ ๐ such that ๐ = ๐๐.
One way to construct it is as follows: Let F be the free module with basis ๐ ร ๐, and let ๐บ be the submodule generated by ๐ฅ + ๐ฅโฒ , ๐ฆ โ ๐ฅ, ๐ฆ โ (๐ฅโฒ , ๐ฆ), ๐ฅ, ๐ฆ + ๐ฆ โฒ โ ๐ฅ, ๐ฆ โ (๐ฅ, ๐ฆ โฒ), ๐ฅ๐, ๐ฆ โ (๐ฅ, ๐๐ฆ) for all ๐ฅ, ๐ฅโฒ โ ๐; ๐ฆ, ๐ฆ โฒ โ ๐, ๐ โ ๐ . Then ๐ โ๐ ๐ = ๐น/๐บ; ๐ฅ โ ๐ฆ denotes ๐ฅ, ๐ฆ + ๐บ. The relations make ๐ฅ โ ๐ฆ biadditive and R-balanced:
1. ๐ฅ + ๐ฅโฒ โ ๐ฆ = ๐ฅ โ ๐ฆ + ๐ฅโฒ โ ๐ฆ 2. ๐ฅ โ ๐ฆ + ๐ฆ โฒ = ๐ฅ โ ๐ฆ + ๐ฅ โ ๐ฆ โฒ
3. ๐ฅ๐ โ ๐ฆ = ๐ฅ โ ๐๐ฆ
If M is a (S,R) bimodule and N is a (R,T) module, then ๐ โ ๐ is a S-T bimodule. Definitions generalize to more modules with multiadditive balanced maps.
The above properties all hold except for commutativity; (2) is replaced by ๐ โ๐ ๐ โ๐ ๐ โ ๐ โ๐ ๐ โ๐ ๐ โ ๐ โ๐ (๐ โ๐ ๐).
๐ ร ๐ ๐
๐ ๐
๐
๐
6 Galois Theory 6-1 Symmetric Polynomials
Symmetric Polynomials A symmetric polynomial is fixed by every permutation of the variables.
The elementary symmetric polynomial in n variables ๐ฅ1, โฆ , ๐ฅ๐ of degree k is
๐ ๐ = ๐ฅ๐ 1 โฏ๐ฅ๐ ๐
1โค๐(1)<โฏ<๐(๐)โค๐
.
Vietaโs Theorem: If ๐ฅ โ ๐ผ1 โฏ ๐ฅ โ ๐ผ๐ = ๐ฅ๐ + ๐1๐ฅ๐โ1 + โฏ + ๐๐
Then ๐ผ๐ = โ1 ๐๐๐ .
Newtonโs identities: Let ๐ค๐ = ๐ผ1๐ + โฏ + ๐ผ๐
๐ . Then
๐ค๐ โ ๐ 1๐ค๐โ1 + โฏ + โ1 ๐๐ ๐๐ค1 + โ1 ๐๐๐ ๐ = 0 Fundamental Theorem of Symmetric Polynomials: Every symmetric polynomial in ๐ [๐ฅ] can be written in a unique way as a polynomial in the elementary symmetric polynomials. The polynomial will be in ๐ [๐ฅ]. Pf. Introduce a lexicographic ordering of the monomials and use induction. Corollaries:
1. If ๐ ๐ฅ โ ๐น[๐ฅ] has roots ๐ผ1, โฆ , ๐ผ๐ in ๐พ โ ๐น, and ๐ ๐ฅ1, โฆ , ๐ฅ๐ โ ๐น[๐ฅ1, โฆ , ๐ฅ๐ ] is a symmetric polynomial, then ๐ ๐ผ1, โฆ , ๐ผ๐ โ ๐น.
2. If ๐1, โฆ , ๐๐ is the orbit of ๐1 for the operation of the symmetric group on the variables, and ๐(๐ฅ1, โฆ , ๐ฅ๐) is a symmetric polynomial, then ๐(๐1, โฆ , ๐๐) is a symmetric polynomial.
6-2 Discriminant If ๐ ๐ฅ = ๐ฅ๐ + ๐1๐ฅ๐โ1 + โฏ + ๐๐ has roots ๐ผ1, โฆ , ๐ผ๐ , then the discriminant of P is
๐ท = ๐ผ๐ โ ๐ผ๐ 2
1โค๐<๐โค๐
.
Since D is a symmetric polynomial in the ๐ข๐ , it can be written in terms of the elementary symmetric polynomials. (It can always be defined in terms of ฮ whether or not P splits completely.)
๐ท ๐ผ1, โฆ , ๐ผ๐ = ฮ ๐ 1, โฆ , ๐ ๐ = ฮ(โ๐1, โฆ , โ1 ๐๐๐) Ex.
1. ๐ท ๐ฅ2 + ๐๐ฅ + ๐ = ๐2 โ 4๐.
2. ๐ท ๐ฅ3 + ๐๐ฅ + ๐ = โ4๐3 โ 27๐2.
6-3 Galois Group Assume fields have characteristic 0. The F-automorphisms of an extension K form the Galois group ๐บ(๐พ/๐น) of K over F. ๐พ/๐น is
a Galois extension3 if |๐บ(๐พ/๐น)| = [๐พ: ๐น]. Ex. ๐บ(โ/โ) = {๐ผ, conjugation}. Any two splitting fields of ๐ over F are isomorphic.
3 In general (when K is not necessarily a finite extension) a Galois extension is defined as a normal, separable field
extension. A separable polynomial has no repeated roots in a splitting field; n element is separable over F if its minimal
polynomial is separable; a separable field extension has every element separable over F.
Pf. (a) An extension field contains at most one splitting field of ๐ over F. (b) Suppose ๐พ1, ๐พ2 are 2 splitting fields. Take a primitive element ๐พ โ ๐พ1 with irreducible polynomial ๐. Choose
an extension L of ๐พ2 so that g has a root ๐พโฒ. Then use (a).
6-4 Fixed Fields
Let H be a group of automorphisms of a field K. The fixed field of H, ๐พ๐ป, is the set of elements of K fixed by every group element.
๐พ๐ป = ๐ผ โ ๐พ ๐ ๐ผ = ๐ผ โ๐ โ ๐ป Fixed Field Theorem:
1. ๐พ: ๐พ๐ป = ๐ป : The degree of K over ๐พ๐ป is equal to the order of the group. 2. ๐ป = ๐บ(๐พ/๐พ๐ป): K is a Galois extension of ๐พ๐ป with Galois group H.
Pf.
1. Let ๐ฝ1 โ ๐พ with H-orbit ๐ฝ1, โฆ , ๐ฝ๐ . For any ๐, there is ๐ โ ๐ป with ๐ ๐ฝ1 = ๐ฝ๐ . Thus ๐ฅ โ ๐ฝ๐ |๐. Since ๐ ๐ฅ = ๐ฅ โ ๐ฝ1 โฏ ๐ฅ โ ๐ฝ๐ โ ๐พ๐ป[๐ฅ] by symmetry, ๐ is the irreducible
polynomial for ๐ฝ1 over ๐พ๐ป. r divides the order of H. 2. If ๐พ: ๐น = โ, there exist elements in K whose degrees over F are arbitrarily large.
3. By (1), ๐พ/๐พ๐ป is algebraic, so by (2), [๐พ: ๐พ๐ป] is finite. The stabilizer of a primitive
element is trivial, so the orbit has order ๐ = |๐ป|. By (1), ๐พ has degree ๐ over ๐พ๐ป.
Then ๐พ: ๐พ๐ป = ๐. 4. Let ๐บ = ๐บ(๐พ/๐พ๐ป). Then ๐ป โ ๐บ โ ๐พ๐บ โ ๐พ๐ป. By definition, every element of G acts as
the identity on ๐พ๐ป so ๐พ๐ป โ ๐พ๐บ . Lรผrothโs Theorem: Let ๐น โ โ be a subfield of the field โ(๐ก) of rational functions. Then ๐น is a field โ(๐ข) of rational functions.
6-5 Galois Extensions and Splitting Fields Splitting Fields A splitting field of ๐ โ ๐น[๐ฅ] over ๐น is an extension ๐พ/๐น such that
1. ๐ splits completely in K: ๐ ๐ฅ = ๐ฅ โ ๐ผ1 โฏ ๐ฅ โ ๐ผ๐ , ๐ผ๐ โ ๐พ. 2. ๐พ = ๐น(๐ผ1, โฆ , ๐๐)
A splitting field is a finite extension, and every finite extension is contained in a splitting field. Splitting Theorem: If K is the splitting field of some ๐ ๐ฅ โ ๐น[๐ฅ], then any irreducible
polynomial ๐ ๐ฅ โ ๐น[๐ฅ] with one root in K splits completely in K. (A field satisfying the latter condition is called a normal extension of F.) Conversely, any finite normal extension is a splitting field. Pf. Suppose ๐(๐ฅ) has the root ๐ฝ โ ๐พ. Then ๐1 ๐ผ1, โฆ , ๐ผ๐ = ๐ฝ for some ๐1 โ ๐น[๐ฅ1, โฆ , ๐ฅ๐]. Let
๐1, โฆ , ๐๐ be the orbit of ๐1 under the symmetric group. Then โ ๐ฅ โ ๐๐ ๐ผ1, โฆ , ๐ผ๐ ๐๐=1 โ ๐น[๐ฅ]
by symmetry so it is divisible by ๐(๐ฅ), the irreducible polynomial of ๐ฝ.
If K is an extension of F, then an intermediate field satisfies ๐น โ ๐ฟ โ ๐พ. A proper intermediate field is neither F nor K. Note ๐น โ ๐ฟ โ ๐บ ๐พ ๐ฟ โ ๐บ(๐พ ๐น ).
The order of ๐บ = ๐บ(๐พ ๐น ) divides [๐พ: ๐น], since ๐บ = [๐พ: ๐พ๐บ] and ๐พ: ๐น = ๐พ: ๐พ๐บ [๐พ๐บ : ๐น]. Characteristic Properties of Galois Extensions: For a finite extension K, the following are equivalent.
1. ๐พ ๐น is a Galois extension. 2. ๐พ๐บ = ๐น where ๐บ = ๐บ(๐พ ๐น ).
3. ๐พ is a splitting field over F. Pf. (1)โ(2): By the Fixed Field Theorem, ๐บ = [๐พ: ๐พ๐บ]. (1)โ(3): Let ๐พ1 be a primitive element for K, with irreducible polynomial ๐. Let ๐พ1, โฆ , ๐พ๐ be
the roots of ๐ in K. There is a unique F-automorphism ๐๐ sending ๐พ1 โ ๐พ๐ for each i, and these make up the group ๐บ(๐พ ๐น ). Thus the order of ๐บ(๐พ ๐น ) is equal to the number of
conjugates of ๐พ1 in K. So ๐พ ๐น Galoisโ ๐ = ๐บ = ๐พ: ๐พ๐บ โ ๐ (degree r) splits completely in
Kโ K is a splitting field. If ๐พ ๐น is a Galois extension, and ๐ โ ๐น[๐ฅ] splits completely in K with roots ๐ฝ1, โฆ , ๐ฝ๐ , then
G operates on the set of roots {๐ฝ๐}. G operates faithfully if K is a splitting field of ๐ over ๐น.
G operates transitively if ๐ is irreducible over F.
If K is the splitting field of irreducible ๐, then G embeds as a transitive subgroup of ๐๐ .
6-6 Fundamental Theorem Fundamental Theorem of Galois Theory: Let K be a Galois extension of a field F, and let ๐บ = ๐บ(๐พ/๐น). Then there is a bijection between subgroups of G and intermediate fields, defined by
๐ป โ ๐พ๐ป ๐บ(๐พ ๐ฟ ) โ ๐ฟ
Let ๐ฟ = ๐พ๐ป (the fixed field of a subgroup H of G). ๐ฟ ๐น is a Galois extension iff H is a normal subgroup of G. If so, ๐บ ๐ฟ ๐น โ ๐บ ๐ป .
Pf. Let ๐พ1 be a primitive element for ๐ฟ ๐น and let ๐ be the irreducible polynomial for ๐พ1 over
F. Let the roots of ๐ in ๐พ be ๐พ1, โฆ , ๐พ๐ . For ๐ โ ๐บ, ๐ ๐พ1 = ๐พ๐ , the stabilizer of ๐พ๐ is ๐๐ป๐โ1.
Thus ๐๐ป๐โ1 = ๐ป โ ๐พ๐ โ ๐ฟ = ๐พ๐ป. H is normalโAll ๐พ๐ โ ๐ฟ โ ๐ฟ ๐น Galois. Restricting ๐ to L gives a homomorphism ๐: ๐บ โ ๐บ(๐ฟ ๐น ) with kernel H.
6-7 Roots of Unity
Let ๐๐ = ๐2๐๐
๐ . ๐น(๐๐) is a cyclotomic field. For p prime, the Galois group of ๐น(๐๐) is
isomorphic to ๐ฝ๐ร, of order ๐ โ 1.
Ex. For ๐ = 2๐ + 1, G is cyclic of order 2๐ . Let ๐ be a primitive root modulo ๐, and ๐ ๐ = ๐๐ .
The degree of each extension in the following chain is 2: ๐น = ๐พโ๐โ โ ๐พโ๐2โ โ โฏ โ ๐พโ๐2๐โ1โ โ
๐พโ๐2๐โ = ๐พ. cos
2๐
๐ generates ๐พโ๐2๐โ1
โ so the regular p-gon can be constructed.
Kronecker-Weber Theorem: Every Galois extension of โ whose Galois group is abelian is contained in a cyclotomic field โ(๐๐). Ex. If p is prime and L is the unique quadratic extension of โ in โ(๐๐),
1. If ๐ โก 1 (mod 4) then ๐ฟ = โ( ๐).
2. If ๐ โก 3 (mod 4) then ๐ฟ = โ(๐ ๐).
Show this by letting ๐ be a generator as before, take the orbit sums of the roots for โ๐2โ. Kummer Extensions
K
L
F
{ }
} ๐บ = ๐บ(๐พ ๐น ) operates on K fixing F.
๐ป = ๐บ(๐พ ๐ฟ ) operates on K fixing L.
๐ป normal: ๐บ ๐ป โ ๐บ ๐ฟ ๐น operates here.
Let ๐น be a subfield of โ containing ๐ = ๐2๐๐ ๐ , and let ๐พ ๐น be a Galois extension of degree p. Then K is obtained by adjoining a pth root (some ๐ฝ with ๐ฝ๐ โ ๐น). Pf.
1. For ๐ โ ๐น, ๐ ๐ฅ = ๐ฅ๐ โ ๐ is either irreducible in F, or splits completely in F. (Take
๐ผ โ ๐ โ ๐บ(๐พ ๐น ); then ๐๐ ๐ฝ = ๐๐๐ฃ๐ฝ for some ๐ฃ โ 0, so G operates transitively on the roots of ๐.)
2. K is a vector space over F; each ๐ โ ๐บ is a linear operator. Choose a generator ๐.
๐๐ = ๐ผ implies that the matrix is diagonalizable and all eigenvalues are powers of ๐.
Let ๐ฝ be an eigenvector with eigenvalue ๐ โ 1; then ๐ ๐ฝ๐ = ๐ฝ๐ โ ๐ฝ๐ โ ๐พ๐บ = ๐น, but
๐ฝ โ ๐น, so ๐น ๐ฝ = ๐พ.
6-8 Cubic Equations Let K be the splitting field of an irreducible cubic polynomial ๐ over F with roots ๐ผ1, ๐ผ2 , ๐ผ3, let
D be the discriminant of ๐, and let ๐บ = ๐บ ๐พ ๐น . Ifโฆ ๐พ: ๐น ๐บ โ Chain Proper intermediate fields
๐ท is not a square in F
3 ๐ด3 โ ๐ถ3 ๐น โ ๐น ๐ผ1 = ๐พ None
๐ท is a square in F 6 ๐บ โ ๐3 ๐น โ ๐น ๐ผ1 โ ๐น ๐ผ1, ๐2 = ๐พ
๐น ๐ผ1 , ๐น ๐ผ2 , ๐น ๐ผ3 , ๐น(๐ฟ)
Pf. Let ๐ฟ = ๐ผ1 โ ๐ผ2 ๐ผ1 โ ๐ผ3 ๐ผ2 โ ๐ผ3 = ยฑ ๐ท. ๐ฟ โ ๐น โ ๐ฟ fixed by every element of G โ only even permutations in G.
In general, for an irreducible polynomial of any degree, ๐ฟ = ๐ท โ ๐น โ G contains only even permutations.
Cubic Formula
To solve ๐ฅ3 + ๐2๐ฅ2 + ๐1๐ฅ + ๐0 = 0 first substitute ๐ฅ = ๐ฆ โ ๐2/3 (Tschirnhausen
transformation) to put it in the form ๐ฅ3 + ๐๐ฅ + ๐ = 0.
For roots ๐ผ1, โฆ , ๐ผ๐ , ๐ง = ๐ผ1 + ๐๐ผ2 + ๐2๐ผ3, ๐งโฒ = ๐ผ1 + ๐2๐ผ2 + ๐๐3 are eigenvectors for
๐ = (123). Let ๐ด = ๐ผ13
cyc , ๐ต = ๐ผ12
cyc ๐ผ2, ๐ถ = ๐ผ1๐ผ22
cyc . Then ๐ต โ ๐ถ = ๐ท. Express
๐ด, ๐ต + ๐ถ in terms of elementary symmetric polynomials, apply Vietaโs formula, and solve for ๐ด, ๐ต, ๐ถ. Find ๐ง3, then take the cube root:
๐ฅ = โ๐
2+
๐3
27+
๐2
4
3
โ ๐
2+
๐3
27+
๐2
4
3
6-9 Quartic Equations Let ๐ โ ๐น[๐ฅ] be an irreducible quartic polynomial.
Let ๐ฝ1 = ๐ผ1๐ผ2 + ๐ผ3๐ผ4, ๐ฝ2 = ๐ผ1๐ผ3 + ๐ผ2๐ผ4, ๐ฝ3 = ๐ผ1๐ผ4 + ๐ผ2๐ผ3. ๐ ๐ฅ = ๐ฅ โ ๐ฝ1 ๐ฅ โ ๐ฝ2 (๐ฅ โ ๐ฝ3) is the resolvent cubic of ๐.
What is ๐บ = ๐บ(๐พ/๐น)?
D square D not square
๐ reducible ๐ท2 (๐ splits completely) ๐ท4 or ๐ถ4 (๐ has 1 root in F)
๐ irreducible ๐ด4 ๐4
๐ท2 = {๐ผ, 12 34 , 13 24 , 14 23 } For the ambiguous case, let ๐พ = ๐ผ1๐ผ2 โ ๐ผ3๐ผ4, ๐ = ๐ผ1 + ๐ผ2 โ ๐ผ3 โ ๐ผ4. ๐ฟ๐พ or ๐ฟ๐ is a square in F iff ๐บ = ๐ถ4.
Pf. The ๐ฝ๐ are distinct since ๐ฝ1 โ ๐ฝ2 = ๐ผ1 โ ๐ผ4 (๐ผ2 โ ๐ผ3). ๐4 operates on ๐ต = {๐ฝ๐}, giving ๐: ๐4 โ ๐3. If ๐ irreducible, G operates transitively on B, so 3| ๐บ .
Special case: ๐ ๐ฅ = ๐ฅ4 + ๐๐ฅ2 + ๐ = 0. Then the roots are ๐ผ1, ๐ผ2 = โ๐ยฑ ๐2โ4๐
2,
๐ผ3 = โ๐ผ1, and ๐ผ4 = โ๐ผ2, so ๐บ โ ๐ท4. Look at expressions such as ๐ผ1๐ผ2. Quartics are solvable in the following way:
1. Adjoin ๐ฟ = ๐ท. 2. Use Cardanoโs formula to solve for a root of the resolvent cubic ๐(๐ฅ) and adjoin it.
3. The Galois group over the field extension K is a subgroup of ๐ท2. At most 2 more square root extensions suffice.
6-10 Quintic Equations and the Impossibility Theorem Quintic Equations Impossibility Theorem: The general quintic equation is not solvable by radicals. Pf.
1. The following are equivalent: (We say that ๐ผ is solvable [by radicals] over F.) a. There is a chain of subfields ๐น = ๐น0 โ ๐น1 โ โฏ โ ๐น๐ = ๐พ โ โ such that ๐ผ โ
๐น๐ , ๐น๐+1 = ๐น๐ (๐ฝ๐ +1) where a power of ๐ฝ๐ +1 is in ๐น๐ .
b. There is a chain of subfields ๐น = ๐น0 โ ๐น1 โ โฏ โ ๐น๐ = ๐พ โ โ such that ๐ผ โ ๐น๐ , and ๐น๐+1 is a Galois extension of ๐น๐ of prime degree. (Equivalent to (a) by
Kummerโs Theorem.)
c. There is a chain of subfields ๐น = ๐น0 โ ๐น1 โ โฏ โ ๐น๐ = ๐พ โ โ such that ๐ผ โ ๐น๐ , and ๐น๐+1 is an abelian Galois extension of ๐น๐ .
2. Let ๐, ๐ โ ๐น[๐ฅ], and let ๐นโฒ be a splitting field of ๐๐. ๐พโฒ contains a splitting field ๐พ of ๐, and a splitting field ๐นโฒ of ๐. Let ๐บ = ๐บ ๐พ ๐น , ๐ป = ๐บ ๐นโฒ ๐น , ๐ข = ๐บ(๐พโฒ /๐น). Then ๐บ, ๐ป
are quotients of ๐ข (Fundamental Theorem) and ๐ข is isomorphic to a subgroup of
๐บ ร ๐ป.
a. Let the canonical maps ๐ข โ ๐บ, ๐ข โ ๐ป send ๐ โ ๐๐ , ๐ โ ๐๐ . Then ๐ข โ ๐บ ร ๐ป
defined by ๐ โ (๐๐ , ๐๐) is injective.
3. Let ๐ โ ๐น[๐ฅ] be a polynomial whose Galois group G is simple and nonabelian. Let ๐นโฒ be a Galois extension of F with Galois group of prime order, and let ๐พโฒ be a splitting field of ๐ over ๐นโฒ. Then ๐บ ๐พโฒ ๐นโฒ โ ๐บ. In other words, we cannot make progress solving for the roots by replacing F by a prime extension.
a. From (3), |๐บ| divides |๐ข| and |๐ข| divides ๐บ ร ๐ป = ๐|๐บ|. 2 cases:
i. ๐ข = ๐บ : Then ๐พโฒ = |๐พ|, ๐ป is a quotient of |๐ข|, contradicting simplicity. ii. ๐ข = ๐บ ร ๐ป : Then ๐บ = ๐บ(๐พโฒ ๐นโฒ ).
4. If ๐ is an irreducible quintic polynomial with Galois group ๐ด5 or ๐5 then the roots of ๐ are not solvable over F.
a. Adjoin ๐ฟ = ๐ท to reduce to ๐ด5 case. b. ๐ด5 is simple. c. By (3), in the โsolvable series,โ each field extension has Galois group ๐ด5. ๐
remains irreducible after each extension.
5. There exists a polynomial with Galois group ๐5.
๐พโฒ
๐นโฒ ๐พ
๐น ๐บ ๐ป
๐ข
a. A subgroup of ๐5 containing a 5-cycle and transposition is the whole group. b. If ๐ is a quintic irreducible polynomial with exactly 3 real roots, then the Galois
group is ๐5 by (a).
c. Example: ๐ฅ5 + 16๐ฅ + 2. In general, a polynomial equation over a field of characteristic 0 is solvable by radicals iff its Galois group is a solvable group. In (4) above, the Galois group after adjoining an element is a subgroup of the previous one, with factor group prime cyclic.
6-11 Transcendence Theory
LindemannโWeierstrass Theorem: If ๐ผ1, . . . , ๐ผ๐ are algebraic numbers linearly independent
over the rational numbers โ, then ๐๐ผ1 , . . . , ๐๐ผ๐ are algebraically independent over โ; in other words the extension field โ ๐๐ผ1 , . . . , ๐๐ผ๐ has transcendence degree n over โ. [Incomplete]
7 Algebras An algebra ๐ over a ring R is a module ๐ over R with multiplication defined so that for all
๐ฅ, ๐ฆ, ๐ง โ ๐, ๐ โ ๐ ,
1. Associative ๐ฅ ๐ฆ๐ง = ๐ฅ๐ฆ ๐ง
2. Distributive ๐ฅ ๐ฆ + ๐ง = ๐ฅ๐ฆ + ๐ฅ๐ง, ๐ฅ + ๐ฆ ๐ง = ๐ฅ๐ง + ๐ฆ๐ง 3. ๐ ๐ฅ๐ฆ = ๐๐ฅ ๐ฆ = ๐ฅ(๐๐ฆ)
If there is an element 1 โ ๐ so that 1๐ฅ = ๐ฅ1 = ๐ฅ, then 1 is the identity element. ๐ is commutative if ๐ฅ๐ฆ = ๐ฆ๐ฅ. ๐ is a division algebra if each element has an inverse (๐ฅ๐ฅโ1 = 1).
7-1 Division Algebras Frobenius Theorem: The only finite-dimensional division algebras D over the real numbers
are โ (the real numbers), โ (the complex numbers) and ๐ป (the quaternions). Pf. Associate with each ๐ โ ๐ท the linear transformation ๐๐ ๐ฅ = ๐๐ฅ.
1. Lemma: The set V of all ๐ โ ๐ท such that ๐2 โ โ, ๐2 โค 0 forms a subspace of codimension 1 (i.e. dimโ(๐/๐ท) = 1).
Proof: Let ๐ be the characteristic polynomial of ๐๐ . By Cayley-Hamilton, ๐ ๐๐ = 0.
Since there are no zero factors in D, one irreducible real factor of ๐ must be 0 at ๐. If the factor is linear (๐ฅ โ ๐), then ๐ โ โ, ๐ = 0. If the factor is irreducible quadratic (๐ฅ2 โ 2โ ๐ ๐ฅ + ๐ 2), then this is the minimal polynomial. Since the minimal
polynomial has the same factors as the characteristic polynomial, ๐ ๐ฅ = ๐ฅ2 โ2โ ๐ ๐ฅ + ๐ 2 ๐ . Since our minimal polynomial has no repeated complex root, ๐๐ is
diagonalizable over โ (see Linear Algebra notes, 8-2). a. If trace ๐๐ = 0, then the eigenvalues are pure imaginary ยฑ๐๐, and ๐๐
2 is
diagonalizable with eigenvalues โ ๐2. Thus ๐๐2 = โ๐2๐ผ๐ โ ๐2 = โ๐2 โค 0.
b. Conversely, if ๐2 โค 0, then ๐๐2 = โ๐2๐ผ๐ for some ๐, and the eigenvalues of ๐๐
can only be ยฑ๐๐. Then trace ๐๐ = 0. c. Hence ๐ = {๐| trace ๐๐ โค 0}. ๐: ๐ โ trace(๐๐) is a linear operator with range
โ (dimension 1). Since V is the kernel, V has codimension 1. 2. From (1), ๐ท = ๐ โ โ. Since ๐2 = {๐ฃ2|๐ฃ โ ๐} gives the reals in (โโ, 0], ๐4 gives the
reals in [0, โ) and V generates D as an algebra.
3. Let ๐ต ๐, ๐ =โ๐๐โ๐๐
2=
๐2+๐2โ ๐+๐ 2
2. From the definition of V, B is an inner product
(positive definite, symmetric). Choose a minimal subspace W of V generating D as
an algebra, and let ๐๐ 1 โค ๐ โค ๐ be an orthonormal basis with respect to B. Then
i โ ๐๐2 = 1, ii ๐๐๐๐ = โ๐๐๐๐(๐ โ ๐).
a. If ๐ = 0, ๐ โ โ.
b. If ๐ = 1, ๐ โ โ. (๐12 = โ1)
c. If ๐ = 2, ๐ โ ๐ป. (Check the relations.)
d. If ๐ > 2, then using then from (ii) ๐1๐2๐๐ 2 = 1 โ ๐1๐2๐๐ = ยฑ1 โ ๐๐ = ยฑ๐1๐2. So span ๐๐ 1 โค ๐ โค ๐ โ 1 โ ๐ with basis ๐๐ 1 โค ๐ โค ๐ โ 1 also generates D as an algebra, contradicting minimality.
Simple and Semisimple Algebras
[See group theory notes.] [Add some stuff here.]
References [A] Algebra by Michael Artin [EoAA] Elements of Abstract Algebra by Richard Dean [PftB] Proofs from the Book by Titu Andreescu and Gabriel Dospinescu [TBGY] Abstract Algebra: The Basic Graduate Year, by Robert Ash Wikipedia