Abstract Algebra - Polynomial

Definition

Polynomial

Let \(A\) be a commutative ring with unity, and \(**x\) an arbitrary symbol**. Every expression of the form

\[ a(x) = a_0 + a_1x + a_2x^2 + \cdots + a_nx^n \]

is called a polynomial in \(x\) with coefficients in \(A\), or more simply, a polynomial in \(x\) over \(A\). The expressions \(a_kx^k\), for \(k \in \{1, \dots, n\}\), are called the terms of the polynomial.

Zero polynomial

The polynomial \(0 + 0x + 0x^2 + \cdots\) of whose coefficients are equal to zero is called the zero polynomial, and is symbolized by \(0\).

Polynomial ring

Let \(A\) be a ring, the symbol \(A[x]\) designates the set of all the polynomials in \(x\) whose coefficients are in \(A\), with polynomial addition and multiplication.

Theorem 1

Let \(A\) be a commutative ring with unity. Then \(A[x]\) is a commutative ring with unity.

Theorem 2

If \(A\) is an integral domain, then \(A[x]\) is an integral domain.

💡 If \(F\) is a field, \(F[x]\) is not necessarily a field.

Theorem 3: Division algorithm for polynomials

If \(a(x)\) and \(b(x)\) arer polynomials over a field \(F\), and \(b(x) \neq 0\), there exist polynomials \(q(x)\) and \(r(x)\) over \(F\) such that

\[ a(x) = b(x)q(x) + r(x), \text{ and } r(x) = 0 \text{ or } \deg r(x) < \deg b(x) \]

Factoring polynomials

Just as every integer can be factored into primes, so every polynomial can be factored into "irreducible" polynomials which cannot be factored further.

Theorem 1

Every ideal of \(F[x]\) is principal.

Theorem 2

Any two nonzero polynomials \(a(x)\) and \(b(x)\) in \(F[x]\) have a gcd \(d(x)\). Furthermore, \(d(x)\) can be expressed as a "linear combination"

\[ d(x) = r(x)a(x) + s(x)b(x) \]

where \(r(x)\) and \(s(x)\) are in \(F[x]\).

Theorem 3: Factorization into irreducible polynomials

Every polynomial \(a(x)\) of positive degree in \(F[x]\) can be written as a product

\[ a(x) = kp_1(x)p_2(x)\cdots p_r(x) \]

where \(k\) is a constant in \(F\) and \(p_1(x), \dots , p_r(x)\) are monic irreducible polynomials of \(F[x]\). More importantly, the factorization is unique.

Substitution of Polynomials

If \(a(x)\) is a polynomial over a field \(F\), say

\[ a(x) = a_0 + a_1x + \cdots + a_nx^n \]

this means that the coefficients \(a_0, a_1, \dots , a_n\) are elements of the field \(F\), while the letter \(x\) is a placeholder which plays no other role than to occupy a given posititon.

If \(a(x)\) is a polynomial over \(F\) and if \(c \in F\), then

\[ a_0 + a_1c + \cdots + a_nc^n \]

is also an element of \(F\), obtatined by substituting \(c\) for \(x\) in the polynomial \(a(x)\). This element is denoted by \(a(c)\). We may regard \(a(x)\) as a function from \(F\) to \(F\).

💡 But the difference between a polynomial and a polynomial function is mainly a difference of viewpoint.

Root

If \(a(x) \in F[x]\) and \(c \in F\) such that \(a(c) = 0\), then we call \(c\) a root of \(a(x)\).

Theorem 1

\(c\) is a root of \(a(x)\) iff \(x - c\) is a factor of \(a(x)\).

Theorem 2

If \(a(x)\) has distinct roots \(c_1, c_2, \dots, c_m\) in \(F\), then \((x-c_1)(x-c_2) \cdots (x - c_m)\) is a factor of \(a(x)\).

Theorem 3

If \(a(x)\) has degree \(n\), it has at most \(n\) roots.

Theorem 4

If \(s/t\) is a root of \(a(x)\), then \(s|a_0\) and \(t|a_n\).

Eisenstein's irreducibility criterion

Let

\[ a(x) = a_0 + a_1x + \cdots + a_nx^n \]

be a polynomial with integer coefficients. Suppose there is a prime number \(p\) which divides every coefficient of \(a(x)\) except the leading coefficient \(a_n\); suppose \(p\) does not divide \(a_n\) and \(p^2\) does not divide \(a_0\). Then \(a(x)\) is irreducible over \(\mathbb{Q}\).

Fundamental Theorem of Algebra

Every non-constant polynomial with complex coefficients has a complex root.

💡 Every polynomial in \(\mathbb{C}[x]\) can be factored into irreducibles. Since.Since the irreducible polynomials are all of degree 1, it follows that if \(a(x)\) is a polynomial of degree \(n\) over \(\mathbb{C}\), it can be factored into \(a(x)=k(x-c_1)(x-c_2)\cdots (x-c_n)\).

Suppose \(a(x) \in \mathbb{R}[x]\). If \(a+bi\) is a root of \(a(x)\), so is \(a-bi\).