Chapter 17 - Polynomial Rings
Section 17.1 - Polynomial Rings
Throughout this chapter, we will assume that \(R\) is a commutative ring with identity.
Definition. Any expression of the form
where \(a_i \in R\) and \(a_n \neq 0\) is called a polynomial over \(R\) with indeterminate \(x\). The elements \(a_0, a_1, \ldots, a_n\) are the coefficients of \(f\). The coefficient \(a_n\) is the leading coefficient.
Definition. A polynomial is known as monic if the leading coefficient is equal to \(1\).
Definition. The degree of \(f\) is the largest nonnegative number such that \(a_n \neq 0\), written as \(\deg f(x) = n\). If no such number exists, that is, \(f(x) = 0\), we say the degree of \(f\) is $-\infty%.
Definition. We denote the set of all polynomials with coefficients in \(R\) as \(R[x]\).
Two polynomials are equal if and only if their corresponding coefficients are equal. When combined with standard addition and multiplication, \(R[x]\) forms a ring.
Theorem. If \(R\) is commutative and has identity, so does \(R[x]\).
Definition. The ring of polynomials with \(n\) indeterminates and coefficients in \(R\) is defined as \(R[x_1][x_2][\ldots][x_n] = R[x_1, x_2, \ldots, x_n]\).
Definition. The evaluation homomorphism is the homomorphism \(\varphi: R[x] \rightarrow R\) defined as \(\varphi(p(x)) = p(\alpha)\) for some \(\alpha \in R\).
Section 17.2 - The Division Algorithm
Theorem. Given \(f(x), g(x) \in F[x]\), where \(F\) is a field and \(g(x) \neq 0\), there exist unique polynomials \(q(x), r(x) \in F[x]\) such that
where either \(\deg r(x) < \deg g(x)\) or \(r(x)\) is the zero polynomial.
Corollary. Let \(F\) be a field. Then, an element \(\alpha \in F\) is a zero of \(p(x) \ in F[x]\) if and only if \((x-\alpha)\) is a factor of \(p(x)\).
Corollary. Let \(F\) be a field. Then, a nonzero polynomial \(p(x) \in F[x]\) with degree \(n\) can have at most \(n\) distinct zeros in \(F\).
Definition. A monic polynomial \(d(x)\) is the greatest common divisor of polynomials \(p(x), q(x) \in F[x]\) if \(d(x)\) evenly divides both \(p(x)\) and \(q(x)\). We write \(\gcd(p(x), q(x)) = d(x)\). This polynomial is unique.
Definition. Two polynomials are relatively prime if their greatest common divisor is \(1\).
Section 17.3 Irreducible Polynomials
Definition. A non-constant polynomial \(f(x) \ in F[x]\) is irreducible over a field \(F\) if it cannot be expressed as the product of two non-identity polynomials \(g(x)\) and \(h(x)\) in \(F[x]\), with the degree of both polynomials strictly less than the degree of \(f(x)\).
Lemma. Let \(p(x) \in \mathbb{Q}[x]\). Then, with \(r, s \in \mathbb{Z}, a(x) \in \mathbb{N}[x]\), we can write \(p(x) = \frac{r}{s} a(x)\).
Lemma. Gauss's Lemma. Let \(p(x) \in \mathbb{Z}[x]\) be monic such that \(p(x)\) factors into two polynomials \(\alpha(x), \beta{x} \in \mathbb{Q}[x]\), with the degrees of both strictly less than the degree of \(p(x)\). Then, there exists two polynomials \(a(x), b(x) \in \mathbb{Z}[x]\) such that \(p(x) = a(x)b(x)\), and \(\deg \alpha(x) = \deg a(x)\) and \(\deg \beta(x) = \deg b(x)\).
Corollary. Let \(p(x) \in \mathbb{Z}[x]\) be monic with constant term \(a_0\). Then, if \(p(x)\) has a zero in \(\mathbb{Q}\), then it also has a zero \(\alpha\) in \(\mathbb[Z]\). Furthermore, \(\alpha\) divides \(a_0\).
Theorem. Eisenstein's Criterion. Let \(p\) be prime, and suppose that
Then, if \(p | a_i\) for \(0 \leq i < n\), but \(p \nmid a_n\) and \(p^2 \nmid a_0\), then \(f(x)\) is irreducible over \(\mathbb{Q}[x]\).
Theorem. If \(F\) is a field, then every ideal in \(F[x]\) is a principal ideal.
Theorem. Let \(F\) be a field, and suppose \(p(x) \in F[x]\). Then, the ideal \(<p(x)>\) is maximal if and only if \(p(x)\) is irreducible.