§5.2 Exercises
Exercises 5.2.A
In Exercises 1–4, write out the addition and multiplication tables for the congruence-class ring \(F[x]/(p(x))\). In each case, is \(F[x]/(p(x))\) a field?
Exercise 5.2.1
\(F = \mathbb{Z}_2\); \(p(x) = x^3 + x + 1\)
Exercise 5.2.2
\(F = \mathbb{Z}_3\); \(p(x) = x^2 + 1\)
Exercise 5.2.3
\(F = \mathbb{Z}_2\); \(p(x) = x^2 + 1\)
Exercise 5.2.4
\(F = \mathbb{Z}_5\); \(p(x) = x^2 + 1\)
Exercises 5.2.B
In Exercises 5–8, each element of the given congruence-class ring can be written in the form \([ax + b]\) (Why?). Determine the rules for addition and multiplication of congruence classes. (In other words, if the product \([ax + b][cx + d]\) is the class \([rx + s]\), describe how to find \(r\) and \(s\) from \(a\), \(b\), \(c\), \(d\), and similarly for addition.)
Exercise 5.2.5
\(\mathbb{R}[x]/(x^2 + 1)\) [Hint: See Example 1.]
Exercise 5.2.6
\(\mathbb{Q}[x]/(x^2 - 2)\)
Exercise 5.2.7
\(\mathbb{Q}[x]/(x^2 - 3)\)
Exercise 5.2.8
\(\mathbb{Q}[x]/(x^2)\)
Exercise 5.2.9
Show that \(\mathbb{R}[x]/(x^2 + 1)\) is a field by verifying that every nonzero congruence class \([ax + b]\) is a unit. [Hint: Show that the inverse of \([ax + b]\) is \([cx + d]\), where \(c = -a/(a^2 + b^2)\) and \(d = b/(a^2 + b^2)\].
Exercise 5.2.10
Let \(F\) be a field and \(p(x) \in F[x]\). Prove that \(F^* = \{[a] \mid a \in F\}\) is a subring of \(F[x]/(p(x))\).
Exercise 5.2.11
Show that the ring in Exercise 8 is not a field.
Exercise 5.2.12
Write out a complete proof of Theorem 5.6 (that is, carry over to \(F[x]\) the proof of the analogous facts for \(\mathbb{Z}\)).
Exercise 5.2.13
Prove the first statement of Theorem 5.7.
Exercise 5.2.14
In each part, explain why \([f(x)]\) is a unit in \(F[x]/(p(x))\) and find its inverse. [Hint: To find the inverse, let \(u(x)\) and \(v(x)\) be as in the proof of Theorem 5.9. You may assume that \(u(x) = ax + b\) and \(v(x) = cx + d\). Expanding \(f(x)u(x) + p(x)v(x)\) leads to a system of linear equations in \(a\), \(b\), \(c\), \(d\). Solve it.]
(a) \([f(x)] = [2x - 3] \in \mathbb{Q}[x]/(x^2 - 2)\)
(b) \([f(x)] = [x^2 + x + 1] \in \mathbb{Z}_3[x]/(x^2 + 1)\)
Exercises 5.2.C
Exercise 5.2.15
Find a fourth-degree polynomial in \(\mathbb{Z}_2[x]\) whose roots are the four elements of the field \(\mathbb{Z}_2[x]/(x^2 + x + 1)\), whose tables are given in Example 3. [Hint: The Factor Theorem may be helpful.]
Exercise 5.2.16
Show that \(\mathbb{Q}[x]/(x^2 - 2)\) is a field.