3.8 KiB
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
f(x) = \sum_{i=0}^n a_i x^i = a_0 + a_1x + a_2x^2 + \ldots + a_n x^n
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 mumber 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 \phi: R[x] \rightarrow R defined as \phi(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
f(x) = g(x)q(x) + r(x)
where either \deg r(x) < \deg g(x) or r(x) is the zero polynomial.
Collary. 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).
Collary. 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 nonconstant 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).
Collary. 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
f(x) = a_n x^n + \ldots + a_0 \in \mathbb{Z}[x]
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.