Chapter 25

Let \(a(x)\) and \(b(x)\) be polynomials of positive degree. By the division algorithm, we may divide \(a(x)\) by \(b(x)\) $$ a(x)=b(x) q_{1}(x)+r_{1}(x) $$ Prove that every common divisor of \(a(x)\) and \(b(x)\) is a common divisor of \(b(x)\) and \(r_{1}(x)\). It follows from part 1 that the ged of \(a(x)\) and \(b(x)\) is the same as the ged of \(b(x)\) and \(r_{1}(x)\). This procedure can now be repeated on \(b(x)\) and \(r_{1}(x)\); divide \(b(x)\) by \(r_{1}(x)\) : Next, $$ \begin{aligned} r_{1}(x) &=r_{2}(x) q_{3}(x)+r_{3}(x) \\ & \vdots \end{aligned} $$ Finally, $$ r_{n-1}(x)=r_{n}(x) q_{n+1}(x)+0 $$ In other words, we continue to divide each remainder by the succeeding remainder. Since the remainders continually decrease in degree, there must ultimately be a zero remainder. But we have seen that $$ \operatorname{gcd}[a(x), b(x)]=\operatorname{gcd}\left[b(x), r_{1}(x)\right]=\cdots=\operatorname{gcd}\left[r_{n-1}(x), r_{n}(x)\right] $$ Since \(r_{n}(x)\) is a divisor of \(r_{n-1}(x)\), it must be the ged of \(r_{n}(x)\) and \(r_{n-1}(x)\). Thus, $$ r_{n}(x)=\operatorname{gcd}[a(x), b(x)] $$ This method is called the euclidean algorithm for finding the gcd.

Prove Euclid's lemma for polynomials.

Let \(F\) be a field, and let \(J\) designate any ideal of \(F[x] .\) Prove the following: Any two generators of \(J\) are associates.

Let \(F\) be a field. Explain why each of the following is true in \(F[x]\). Every polynomial of degree 1 is irreducible.

Factor \(x^{4}-4\) into irreducible factors over \(\mathbf{Q}\), over \(\mathbb{R}\), and over \(\mathbb{C}\).

Let \(h: F[x] \rightarrow F[x]\) be defined by: $$ h\left(a_{0}+a_{1} x+\cdots+a_{n} x^{n}\right)=a_{n}+a_{n-1} x+\cdots+a_{0} x^{n} $$ Prove the following: \(h\) is injective and surjective, hence an automorphism of \(F[x]\).

Let \(F\) be a field, and let \(J\) designate any ideal of \(F[x] .\) Prove the following: J has a unique monic generator \(m(x)\). An arbitrary polynomial \(a(x) \in F[x]\) is in \(J\) iff \(m(x) \mid a(x)\)

How many reducible quadratics are there in \(\mathbb{Z}_{5}[x] ?\) How many irreducible quadratics?

Let \(F\) be a field. Explain why each of the following is true in \(F[x]\). If \(a(x)\) and \(b(x)\) are distinct monic polynomials, they cannot be associates.

Factor \(x^{6}-16\) into irreducible factors over \(\mathbb{Q}\), over \(\mathbb{R}\), and over \(\mathbb{C}\).

Let \(h: F[x] \rightarrow F[x]\) be defined by: $$ h\left(a_{0}+a_{1} x+\cdots+a_{n} x^{n}\right)=a_{n}+a_{n-1} x+\cdots+a_{0} x^{n} $$ Prove the following: \(a_{0}+a_{1} x+\cdots+a_{n} x^{n}\) is irreducible iff \(a_{n}+a_{n-1} x+\cdots+a_{0} x^{n}\) is irreducible.

Let \(F\) be a field, and let \(J\) designate any ideal of \(F[x] .\) Prove the following: \(J\) is a prime ideal iff it has an irreducible generator.

Let \(F\) be a field. Explain why each of the following is true in \(F[x]\). Any two distinct irreducible polynomials are relatively prime.

Find all the irreducible polynomials of degree \(\leq 4\) in \(\mathbb{Z}_{2}[x]\).

Let \(a(x)\) and \(b(x)\) be polynomials of positive degree. By the division algorithm, we may divide \(a(x)\) by \(b(x)\) $$ a(x)=b(x) q_{1}(x)+r_{1}(x) $$ Find the gcd of \(x^{3}+x^{2}+x+1\) and \(x^{4}+x^{3}+2 x^{2}+2 x\) in \(\mathbb{Z}_{3}[x]\).

Let \(F\) be a field. Explain why each of the following is true in \(F[x]\). If \(a(x)\) is irreducible, any associate of \(a(x)\) is irreducible.

Let \(F\) be a field. Explain why each of the following is true in \(F[x]\). If \(a(x) \neq 0, a(x)\) cannot be an associate of 0 .

In \(\mathbb{Z}_{6}[x]\), factor each of the following into two polynomials of degree \(1: x, x+2\) \(x+3\). Why is this possible?