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Chapter-1

Prabal Paul

Department of Mathematics

BITS Goa, Goa

1st September 2014

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Polynomial ring

Definition

Let Rbe an arbitrary ring. A polynomial over R is an expressionof the form

f(x) =n

i=0

aixi =a0+a1x+a2x

2 + +anxn

where n is a non-negative integer, the coefficient ai, 0in,are elements ofRand the symbol xdoes not belong toR, called

an indeterminate over R.

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Polynomial ring

Remark

For two polynomial

f(x) =a0+a1x+ +anxn

, g(x) =m

i=0bixi

, without loss ofgenerality, we can assume that m =n. Then the sum is defined by

f(x) +g(x) = (a0+b0) + (a1+b1)x+ + (am+bm)xm.

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Polynomial ring

Remark

For two polynomialf(x) =a0+a1x+ +anx

n, g(x) =m

i=0bixi, The product is

defined by

f(x)g(x) =n+m

i=0

cixi

where ci =

k1+k2=i,0k1n,0k2mak1 bk2 .

Please look into the board for some examples.

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Polynomial ring

Definition

The ring formed by the polynomials over Rwith the aboveoperations is called the polynomial ring over Rand is denoted byR[x].

Definition

Let Rbe a ring. Let f(x) =a0+a1x+ +anxn be a polynomial

with an = 0 in R. Then an is called the leading coefficient and nis called the degree of the polynomial. The degree of the zeropolynomials is defined to be .

Definition

Let Rbe a ring. A polynomial f in R[x] is called monic if theleading co-efficient off is 1.

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Polynomial ring

Definition

The ring formed by the polynomials over Rwith the aboveoperations is called the polynomial ring over Rand is denoted byR[x].

Definition

Let Rbe a ring. Let f(x) =a0+a1x+ +anxn be a polynomial

with an = 0 in R. Then an is called the leading coefficient and nis called the degree of the polynomial. The degree of the zeropolynomials is defined to be .

Definition

Let Rbe a ring. A polynomial f in R[x] is called monic if theleading co-efficient off is 1.

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Polynomial ring

Definition

The ring formed by the polynomials over Rwith the aboveoperations is called the polynomial ring over Rand is denoted byR[x].

Definition

Let Rbe a ring. Let f(x) =a0+a1x+ +anxn be a polynomial

with an = 0 in R. Then an is called the leading coefficient and nis called the degree of the polynomial. The degree of the zeropolynomials is defined to be .

Definition

Let Rbe a ring. A polynomial f in R[x] is called monic if theleading co-efficient off is 1.

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Polynomial ring

Theorem (division algorithm, theorem 1.52)

Let g(= 0)R[x] be a polynomial in F (where F is a field). Thenfor any f in F[x], there exists q(x), r(x)F[x] withdeg(r(x))< deg(g(x)) such that f(x) =q(x)g(x) +r(x).

Please look into the board for some examples.

Self studies: theorem 1.54 (F[x] is a principal ideal domain).Self studies: theorem 1.55 (Euclidian algorithm for F[x]).

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Polynomial ring

Theorem (division algorithm, theorem 1.52)

Let g(= 0)R[x] be a polynomial in F (where F is a field). Thenfor any f in F[x], there exists q(x), r(x)F[x] withdeg(r(x))< deg(g(x)) such that f(x) =q(x)g(x) +r(x).

Please look into the board for some examples.

Self studies: theorem 1.54 (F[x] is a principal ideal domain).Self studies: theorem 1.55 (Euclidian algorithm for F[x]).

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P l i l i

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Polynomial ring

Theorem (division algorithm, theorem 1.52)

Let g(= 0)R[x] be a polynomial in F (where F is a field). Thenfor any f in F[x], there exists q(x), r(x)F[x] withdeg(r(x))< deg(g(x)) such that f(x) =q(x)g(x) +r(x).

Please look into the board for some examples.

Self studies: theorem 1.54 (F[x] is a principal ideal domain).Self studies: theorem 1.55 (Euclidian algorithm for F[x]).

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Polynomial ring

Definition

A polynomial pF[x] (where F is a field) is said to beirreducible over F ifphas positive degree and p=bc withb, cF[x] implies that either borC is a constant polynomial.

Self studies: lemma 1.58 (irreducible and prime elements are samein F[x]).

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Polynomial ring

Definition

A polynomial pF[x] (where F is a field) is said to beirreducible over F ifphas positive degree and p=bc withb, cF[x] implies that either borC is a constant polynomial.

Self studies: lemma 1.58 (irreducible and prime elements are samein F[x]).

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Polynomial ring

Theorem (1.59, Unique factorization)

Any polynomial f F[x](where F is a field) of positive degree canbe written in the form

f =ape11 pe22 pekk

where aF and p1, , pkare distinct monic irreduciblepolynomials in F[x] and e1, , ek are positive integers. Moreover,this factorization is unique apart from the order in which the

factors occur.

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Polynomial ring

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Polynomial ring

Theorem (1.61)

For f F[x], the residue class ring F[x]/(f) is a field if and only iff is irreducible over F .

Self studies: Example 1.62.

Please look into the board for some examples.

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Polynomial ring

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Polynomial ring

Theorem (1.61)

For f F[x], the residue class ring F[x]/(f) is a field if and only iff is irreducible over F .

Self studies: Example 1.62.

Please look into the board for some examples.

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Field extensions

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Field extensions

Definition

A field containing no proper subfields is called a prime field.

Theorem (1.78)The prime subfield of a field F is isomorphic to eitherFp orQaccording as the characteristic of F is a prime p or0.

Self studies: Theorem 1.69.

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Field extensions

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Field extensions

Definition

A field containing no proper subfields is called a prime field.

Theorem (1.78)The prime subfield of a field F is isomorphic to eitherFp orQaccording as the characteristic of F is a prime p or0.

Self studies: Theorem 1.69.

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Thank you

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