0. introduction - math 407: modern algebra icampbell/math407spr08/notes/00_intro/slides.pdf · umbc...
TRANSCRIPT
0. IntroductionMath 407: Modern Algebra I
Robert Campbell
UMBC
January 29, 2008
Robert Campbell (UMBC) 0. Introduction January 29, 2008 1 / 22
Outline
1 Math 407: Abstract Algebra
2 Sources
3 Cast of Characters
4 Background Material
5 Applications & Follow-Up
Robert Campbell (UMBC) 0. Introduction January 29, 2008 2 / 22
UMBC Course Description
The basic abstract algebraic structures (rings, integral domains, divisionrings, fields and Boolean algebra) will be introduced, and the fundamentalconcepts of number theory will be examined from an algebraic perspective.This will be done by examining the construction of the natural numbersfrom the Peano postulates, the construction of the integers from thenatural numbers, the rationals as the field of quotients of the integers, thereals as the ordered field completion of the rationals and the complexnumbers as the algebraic completion of the reals. The basic concepts ofnumber theory lead to modular arithmetic; ideals in rings; and to examplesof integral domains, division rings and fields as quotient rings. The conceptof primes yields the algebraic concepts of unique factorization domains,Euclidean rings, and prime and maximal ideals of rings. Examples ofsymmetries in number theory and geometry lead to the concept of groupswhose fundamental properties and applications will be explored.
Robert Campbell (UMBC) 0. Introduction January 29, 2008 3 / 22
Algebra
Def: Algebra is a branch of mathematics that
utilizes symbols, as letters, to represent specific numbers, values ofvectors. (Webster)
concerns the study of structure, relation and quantity. (Wikipedia)
From the title “Hisab al-jabr w’al-muqabala, Kitab al-Jabrwa-l-Muqabala” (The Compendious Book on Calculation byCompletion and Balancing) by Al-Khwarizmi (circa 820 AD)
Algebra generally refers to polynomial equations (aka algebraic equations)
Robert Campbell (UMBC) 0. Introduction January 29, 2008 4 / 22
Abstract
Def: Abstract
(adj) Thought of apart from concrete realities or specific objects.
(vt) To summarize
cow, cow, cow, dog, dog, dog −→ 3
Loss, debt −→ (−3)
Hypotenuse of isosceles right triangle −→√
2 (irrational)
x2 + 1 = 0 −→ i =√−1
Robert Campbell (UMBC) 0. Introduction January 29, 2008 5 / 22
Outline
1 Math 407: Abstract Algebra
2 Sources
3 Cast of Characters
4 Background Material
5 Applications & Follow-Up
Robert Campbell (UMBC) 0. Introduction January 29, 2008 6 / 22
Number Theory
aka Algebra over the integers
Solve 6x + 15y = 9 (Linear Diophantine equation)
Solve x2 + y2 = z2 (Pythagorean triples)
Solve x3 + 3y2 = 5 (Elliptic curve)
Solve x3 + y3 = z3 (subcase of Fermat’s Last Theorem)
Robert Campbell (UMBC) 0. Introduction January 29, 2008 7 / 22
Geometry
The sides of an isosceles right triangle are incommensurable (√
2 isirrational) [Pythagorus, ca 500 BC]
Angles cannot be trisected (with compass and ruler)
A right triangle whose sides are integers cannot have area which is asquare or twice a square [Conj: Fermat, 165?, unproven]
Robert Campbell (UMBC) 0. Introduction January 29, 2008 8 / 22
Algebra
Solutions to quadratic equations
Solutions to cubic and quartic equations
Can solutions to x5 + ax4 + 1 = 0 be expressed with just roots?
Solutions to systems of linear equations
Solutions to systems of polynomial equations
Robert Campbell (UMBC) 0. Introduction January 29, 2008 9 / 22
Symmetries
Geometries: Groups of Isometries (distance preservingtransformations)
Symmetry Groups
Geometric FiguresTesselation & Crystallographic Groups
Topology
Robert Campbell (UMBC) 0. Introduction January 29, 2008 10 / 22
Outline
1 Math 407: Abstract Algebra
2 Sources
3 Cast of Characters
4 Background Material
5 Applications & Follow-Up
Robert Campbell (UMBC) 0. Introduction January 29, 2008 11 / 22
The Zoo
Groups
Rings
Fields
Vector Spaces & Modules
Algebras
Robert Campbell (UMBC) 0. Introduction January 29, 2008 12 / 22
Groups
Def: A group is a set with a “multiplication” operation, an identityelement (“multiplication” by it has no effect) and inverses.
Integers with Addition: Z+
Modular Integers with Multiplication: Z∗n
Matrix Groups with Multiplication: GLn(R), SL2(Z), etc
Permutation Groups:
Geometric Symmetries: Tesselations, Polygons, Polyhedra
Robert Campbell (UMBC) 0. Introduction January 29, 2008 13 / 22
Rings
Def: A ring is a set with a “multiplication” operation and a commutative“addition” operation, an element which acts like “0” and another whichacts like “1”, and additive inverses.
Z, Q, R and CSquare Matrices: Mn(R), Mn(Z), etc
Polynomials: Z[x ], C[x ], Q[x , y ], etc
Modular Integers: Zn
Algebraic Integers: Z[√−1], Z[
√3], etc
Real Division Rings: R ⊂ C =< 1, i |i2 = −1 >⊂ H =< 1, i , j , k|i2 =j2 = k2 = ijk = −1 >
Robert Campbell (UMBC) 0. Introduction January 29, 2008 14 / 22
Fields
Def: A field is a set with a commutative “multiplication” operation and acommutative “addition” operation, an element which acts like “0” andanother which acts like “1”, additive and multiplicative inverses (exceptfor “0”).
Q, R and CNumber Fields: Q[
√−1], Q[
√3], etc
Finite Fields: Zp, GF (pn) := Zp[x ]/ < p(x) >
Robert Campbell (UMBC) 0. Introduction January 29, 2008 15 / 22
Vector Spaces & Modules
Def:
A vector space V over a field F is a set with commutative additionand scalar multiplication by elements of the field.
A module M over a ring R is a set with commutative addition andscalar multiplication by elements of the ring.
Vector Space: Rn, Cn, Mn(R), R[x ], etc
Module: Zn, Zk1 ⊕ Zk2 ⊕ · · · , etc
Robert Campbell (UMBC) 0. Introduction January 29, 2008 16 / 22
Algebras
Def: An algebra is a ring which is also a vector space.
Square Matrices: Mn(R), etc
Polynomials: C[x ], Q[x , y ], etc
Real Division Algebras: R ⊂ C =< 1, i |i2 = −1 >⊂ H =<1, i , j , k|i2 = j2 = k2 = ijk = −1 >
Robert Campbell (UMBC) 0. Introduction January 29, 2008 17 / 22
Outline
1 Math 407: Abstract Algebra
2 Sources
3 Cast of Characters
4 Background Material
5 Applications & Follow-Up
Robert Campbell (UMBC) 0. Introduction January 29, 2008 18 / 22
Linear Algebra
Matrix multiplication
Non-commutative multiplication (AB 6= BA)
Zero divisors (e.g.
(0 10 0
) (1 00 0
)=
(0 00 0
))
Nilpotent elements (e.g. if N =
(0 10 0
)then N2 = 0)
Idempotent elements (e.g. if A =
(1 00 0
)then A2 = A, but A 6= 1)
Trace and Det of a linear transformation
Robert Campbell (UMBC) 0. Introduction January 29, 2008 19 / 22
Analysis
Logic
Proofs
Set Theory
C and R - Construction and algebraic closure.
Polynomial and rational functions
Robert Campbell (UMBC) 0. Introduction January 29, 2008 20 / 22
Outline
1 Math 407: Abstract Algebra
2 Sources
3 Cast of Characters
4 Background Material
5 Applications & Follow-Up
Robert Campbell (UMBC) 0. Introduction January 29, 2008 21 / 22
Further Topics
Abstract Algebra II (Math 408)
more Rings & Fieldsmore Group TheoryGalois Theory (combining Group & Field Theory)
Number Theory (Math 413)
Algebraic Number Theory
Algebraic Geometry
Algebraic Topology
Differential Geometry
Robert Campbell (UMBC) 0. Introduction January 29, 2008 22 / 22