ยง4.2 Exercises

Note: \(F\) denotes a field.

Exercises 4.2.A

Exercise 4.2.1

If \(f(x) \in F[x]\), show that every nonzero constant polynomial divides \(f(x)\).

Exercise 4.2.2

If \(f(x) = c_n x^n + \dots + c_0\) with \(c_n \neq 0_F\), what is the gcd of \(f(x)\) and \(0\)?

Exercise 4.2.3

If \(a, b \in F\) and \(a \neq b\), show that \(x + a\) and \(x + b\) are relatively prime in \(F[x]\).

Exercise 4.2.4

(a) Let \(f(x), g(x) \in F[x]\). If \(f(x) | g(x)\) and \(g(x) | f(x)\), show that \(f(x) = cg(x)\) for some nonzero \(c \in F\).

(b) If \(f(x)\) and \(g(x)\) in part (a) are monic, show that \(f(x) = g(x)\).

Exercise 4.2.5

The Euclidean Algorithm for finding gcd's is described for integers in Exercise 15 of Section 1.2. The process given there also works for polynomials over a field, with one minor adjustment. For integers, the last nonzero remainder is the gcd. For polynomials, the last nonzero remainder is the common divisor of highest degree, but it may not be monic. In that case, multiply it by the inverse of its leading coefficient to obtain the gcd. Use the Euclidean Algorithm to find the gcd of the given polynomials:

  1. (a) \(x^4 - x^3 - x^2 + 1$ and $x^3 - 1\) in \(\mathbb{Q}[x]\)
  2. (b) \(x^5 + x^4 + 2x^3 - x^2 - x - 2\) and \(x^4 + 2x^3 + 5x^2 + 4x + 4\) in \(\mathbb{Q}[x]\)
  3. (c) \(x^4 + 3x^3 + 2x + 4$ and $x^2 - 1\) in \(\mathbb{Z}_5[x]\)
  4. (d) \(4x^4 + 2x^3 + 6x^2 + 4x + 5\) and \(3x^3 + 5x^2 + 6x\) in \(\mathbb{Z}_7[x]\)

Exercise 4.2.6

Express each of the gcd's in Exercise 5 as a linear combination of the two polynomials.

Exercises 4.2.B

Exercise 4.2.7

Let \(f(x) \in F[x]\) and assume that \(f(x)g(x)\) for every nonconstant \(g(x) \in F[x]\). Show that \(f(x)\) is a constant polynomial. (*Hint:* \(f(x)\) must divide both \(x + 1\) and \(x - 1\).

Exercise 4.2.8

Let \(f(x), g(x) \in F[x]\), not both zero, and let \(d(x)\) be their gcd. If \(h(x)\) is a common divisor of \(f(x)\) and \(g(x)\) of highest possible degree, then prove that \(h(x) = c d(x)\) for some nonzero \(c \in F\).

Exercise 4.2.9

If \(f(x) \neq 0_F\) and \(f(x)\) is relatively prime to \(0_F\), what can be said about \(f(x)\)?

Exercise 4.2.10

Find the gcd of \(x + a + b\) and \(x^3 - 3abx + a^3 + b^3\) in \(\mathbb{Q}[x]\).

Exercise 4.2.11

Fill in the details of the proof of Theorem 4.8.

Exercise 4.2.12

Prove Corollary 4.9.

Exercise 4.2.13

Prove Theorem 4.10.

Exercise 4.2.14

Let \(f(x), g(x), h(x) \in F[x]\), with \(f(x)\) and \(g(x)\) relatively prime. If \(f(x) | h(x)\) and \(g(x) | h(x)\), prove that the gcd of \(f(x)h(x)\) and \(g(x)\) is the same as the gcd of \(h(x)\) and \(g(x)\).

Exercise 4.2.15

Let \(f(x), g(x), h(x) \in F[x]\), with \(f(x)\) and \(g(x)\) relatively prime. If \(f(x) | h(x)\) and \(g(x) | h(x)\), prove that \(h(x) = f(x)g(x)\).

Exercise 4.2.16

Let \(f(x), g(x), h(x) \in F[x]\), with \(f(x)\) and \(g(x)\) relatively prime. Prove that the gcd of \(f(x)h(x)\) and \(g(x)\) is the same as the gcd of \(h(x)\) and \(g(x)\).