ch 16 to 17: Polynomial Ring and Factorization of Polynomials

Let R be a commutative ring. The ring of polynomials over R in the indeterminate x is

the set of formal symbols R[x] = {a_nxⁿ + a_n-1x^n-1 + … + a₁x + a₀ | a_i belonging to R, n is a nonnegative integer}

Two elements a_nxⁿ +a_n-1 x^n-1 + … + a₁x + a₀ and b_mx^m + b_m-1x^m-1 + … + b₁x + b₀ of R[x] are considered equal if and only if

a_i = b_i for all nonnegative integers i (Define a_i = 0 when i > n and b_i = 0 when i > m

Let R be a commutative ring and let f(x) = a_nxⁿ + a_n-1x^n-1 + … + a₁x + a₀ and g(x) = b_mx^m + b_m-1x^m-1 + … + b₁x + b₀ belong to R[x]. Then

f(x) + g(x) =

(a_s + b_s) x^s + (a_s-1 + b_s-1) x^s-1 + … + (a₁ + b₁) x + a₀ + b₀, where s is the maximum of m and n, a_i = 0 for i > n, and b_i = 0 for i > m

Let R be a commutative ring and let f(x) = a_nxⁿ + a_n-1x^n-1 + … + a₁x + a₀ and g(x) = b_mx^m + b_m-1x^m-1 + … + b₁x + b₀ belong to R[x]. Then

f(x) g(x) =

c _m+n x ^m+n + c _m+n-1 x ^m+n-1 + … + c₁x + c₀ , where c_k = a_kb₀ + a_k-1 b₁ + … + a₁ b _k-1 + a₀ b_k for k = 0, … , m + n

Theorem 16.1

If D is an integral domain, then D[x] is

an integral domain

Theorem 16.2 - Division Algorithm for F[x]

Let F be a field and let f(x), g(x) be elements of F[x] with g(x) ≠ 0. Then there exist

unique polynomials q(x) and r(x) in F[x] such that f(x) = g(x) q(x) + r(x) and either r(x) = 0 or deg r(x) < deg g(x)

Corollary 1 - Remainder Theorem

Let F be a field, a be an element of F and f(x) be an element of F[x]. The f(a) is

the remainder in the division of f(x) by x - a

Corollary 2 - Factor Theorem

Let F be a field, a be an element of F, and f(x) be an element of F[x]. Then a is a zero of f(x) if and only if

x - a is a factor of f(x)

Theorem 16.3

A polynomial of degree n over a field has

at most n zeros, counting multiplicity

A principal ideal domain is

an integral domain R in which every ideal has the form < a > = {ra | r belonging to R} for some a in R

Theorem 16.4

Let F be a field. The F[x] is

a principal ideal domain

Theorem 16.5

Let F be a field, I a nonzero ideal in F[x], and g(x) an element of F[x]. Then, I = < g (x) > if and only if

g(x) is a polynomial of minimum degree in I

Let D be an integral domain. A polynomial f(x) from D[x] that is neither the zero polynomial nor a unit in D[x] is said to be irreducible over D if,

whenever f(x) is expressed as a product f(x) = g(x) h(x), with g(x) and h(x) from D[x], then g(x) or h(x) isa unit in D[x]

A nonzero, nonunit element of D[x] that is not irreducible over D is

reducible over D

Theorem 17.1

Let F be a field. If f(x) is an element of F[x] and deg f(x) is 2 or 3, then f(x) is reducible over F if and only if

f(x) has a zero in F

The content of a nonzero polynomial a_n xⁿ + a_n-1 x ^n-1 + … + a₀, where the a’s are integers, is

the greatest common divisor of the integers a_n , a_n-1 , …, a₀

A primitive polynomial is

an element of Z[x] with content 1

Gauss’s Lemma

The product of two primitive polynomials is

primitive

Theorem 17.2

Let f(x) be an element of Z[x]. If f(x) is reducible over Q, then it is

reducible over Z

Theorem 17.3

Let p be a prime and suppose that f(x) be an element of Z[x] with deg f(x) ≥ 1. Let f’(x) be the polynomial in Z_p [x] obtained from f(x) by reducing all the coefficients of f(x) modulo p.

If f’(x) is irreducible over Z_p and deg f’(x) = deg f(x), then f(x) is

irreducible over Q

Theorem 17.4 - Eisenstein’s Criterion

Let f(x) = a_n xⁿ + a_n-1 x^n-1 + … + a₀ be an element of Z[x].

If there is a prime p such that p cannot divide a_n , p can divide a_n-1 , … , p can divide a₀ and p² cannot divide a₀ , then f(x) is

irreducible over Q

Theorem 17.5

Let F be a field and let p(x) be an element of F[x]/ Then < p(x) > is a maximal ideal in F[x] if and only if

p(x) s irreducible over F

Corollary 1

Let F be a field and p(x) be an irreducible polynomial F. The F[x] / < p(x) > is

a field

Corollary 2

Let F be a field and let p(x), a(x), b(x) be an element of F[x]. If p(x) is irreducible over F and p(x) | a(x) b(x), then

p(x) | a(x) or p(x) | b(x)

Theorem 17.6 pt 1

Every polynomial in Z[x] that is not the zero polynomial or a unit in Z[x] can be written in the form

b₁ b₂ … b_s p₁ (x) p₂ (x) … p_m (x), where b_i’s are irreducible polynomials of degree 0 and p_i (x)’s are irreducible polynomials of positive degrees.

Theorem 17.6 pt 2

If b₁ b₂ … b_s p₁ (x) p₂ (x) … p_m (x) = c₁ c₂ … c_t q₁(x) q₂ (x) … q_n (x), where the b_i’s and c_i’s are irreducible polynomials of degree 0 and p_i (x)’s and q_i’s are irreducible polynomials of positive degree, then

s = t, m = n, and , after renumbering the c’s and q(x)’s , we have b_i = ± c_i for i = 1, … , s and p_i (x) = ± q_i (x) for i = 1, …, m