§5.3 Exercises

Exercise 5.3.1

Determine whether the given congruence-class ring is a field. Justify your answer.

(a) \(\mathbb{Z}_3[x]/(x^3 + 2x^2 + x + 1)\)

(b) \(\mathbb{Z}_5[x]/(2x^3 - 4x^2 + 2x + 1)\)

(c) \(\mathbb{Z}_3[x]/(x^4 + x^2 + 1)\)

Exercise 5.3.2

(a) Verify that \(\mathbb{Q}(\sqrt{2}) = \{r + s\sqrt{2} \mid r, s \in \mathbb{Q}\}\) is a subfield of \(\mathbb{R}\).

(b) Show that \(\mathbb{Q}(\sqrt{2})\) is isomorphic to \(\mathbb{Q}[x]/(x^2 - 2)\). [Hint: Exercise 6 in Section 5.2 may be helpful.]

Exercise 5.3.3

If \(a \in F\), describe the field \(F[x]/(x - a)\).

Exercise 5.3.4

Let \(p(x)\) be irreducible in \(F[x]\). Without using Theorem 5.10, prove that if \([f(x)][g(x)] = [0_F]\) in \(F[x]/(p(x))\), then \([f(x)] = [0_F]\) or \([g(x)] = [0_F]\). [Hint: Exercise 10 in Section 5.1.]

Exercise 5.3.5

(a) Verify that \(\mathbb{Q}(\sqrt{3}) = \{r + s\sqrt{3} \mid r, s \in \mathbb{Q}\}\) is a subfield of \(\mathbb{R}\).

(b) Show that \(\mathbb{Q}(\sqrt{3})\) is isomorphic to \(\mathbb{Q}[x]/(x^2 - 3)\).

Exercise 5.3.6

Let \(p(x)\) be irreducible in \(F[x]\). If \([f(x)] \neq [0]\) in \(F[x]/(p(x))\) and \(h(x) \in F[x]\), prove that there exists \(g(x) \in F[x]\) such that \([f(x)][g(x)] = [h(x)]\) in \(F[x]/(p(x))\). [Hint: Theorems 5.10 and Exercise 12(b) in Section 3.2.]

Exercise 5.3.7

If \(f(x) \in F[x]\) has degree \(n\), prove that there exists an extension field \(E\) of \(F\) such that \(f(x) = c_0(x - c_1)(x - c_2) \dots (x - c_n)\) for some (not necessarily distinct) \(c_i \in E\). In other words, \(E\) contains all the roots of \(f(x)\).

Exercise 5.3.8

If \(p(x)\) is an irreducible quadratic polynomial in \(F[x]\), show that \(F[x]/(p(x))\) contains all the roots of \(p(x)\).

Exercise 5.3.9

(a) Show that \(\mathbb{Z}_2[x]/(x^3 + x + 1)\) is a field.

(b) Show that the field \(\mathbb{Z}_2[x]/(x^3 + x + 1)\) contains all three roots of \(x^3 + x + 1\).

Exercise 5.3.10

Show that \(\mathbb{Q}[x]/(x^2 - 2)\) is not isomorphic to \(\mathbb{Q}[x]/(x^2 - 3)\). [Hint: Exercises 2 and 5 may be helpful.]

Exercise 5.3.11

Let \(K\) be a ring that contains \(\mathbb{Z}_8\) as a subring. Show that \(p(x) = 3x^2 + 1 \in \mathbb{Z}_8[x]\) has no roots in \(K\). Thus, Corollary 5.12 may be false if \(\mathbb{Z}_8[x]\) has no roots. [Hint: If \(u \in \mathbb{Z}_8\), then \(0 = 3u^2 + 1 = 0\). Derive a contradiction.]

Exercise 5.3.12

Show that \(2x^3 + 4x^2 + 8 + 3 \in \mathbb{Z}_{16}[x]\) has no roots in any ring \(K\) that contains \(\mathbb{Z}_{16}\) as a subring. [See Exercise 11.]

Exercise 5.3.13

Show that every polynomial of degree 1, 2, or 4 in \(\mathbb{Z}_8[x]\) has a root in \(\mathbb{Z}_8[x]/(x^4 + x + 1)\).