§5.1 Exercises

Exercise 5.1.1

Let \(f(x), g(x), p(x) \in F[x]\), with \(p(x)\) nonzero. Determine whether \(f(x) \equiv g(x) \pmod{p(x)}\). Show your work.

1. \(f(x) = x^5 - 2x^4 + 4x^3 + x + 1\); \(g(x) = 3x^4 + 2x^3 - 5x^2 - 9\); \(p(x) = x^3 + x + 1\); \(F = \mathbb{Q}\)
2. \(f(x) = x^4 + x^2 + x + 1\); \(g(x) = x^4 + 3x^2 + 1\); \(p(x) = x^2 + x\); \(F = \mathbb{Z}_2\)
3. \(f(x) = 3x^5 + 4x^4 + 5x^3 - 6x^2 + 5x - 7\); \(g(x) = 2x^5 + 6x^4 + 3x^2 + 2x - 5\); \(p(x) = x^3 - x^2 + x - 1\); \(F = \mathbb{R}\)

Exercise 5.1.2

If \(p(x)\) is a nonzero constant polynomial in \(F[x]\), show that any two polynomials in \(F[x]\) are congruent modulo \(p(x)\).

Exercise 5.1.3

How many distinct congruence classes are there modulo \(x^3 + 2x + 1\) in \(\mathbb{Z}_3[x]\)? List them.

Exercise 5.1.4

Show that, under congruence modulo \(x^3 + 2x + 1\) in \(\mathbb{Z}_3[x]\), there are exactly 27 distinct congruence classes.

Exercise 5.1.5

Show that there are infinitely many distinct congruence classes modulo \(x^2 - 2\) in \(\mathbb{Q}[x]\). Describe them.

Exercise 5.1.6

Let \(a \in F\). Describe the congruence classes in \(F[x]\) modulo the polynomial \(x - a\).

Exercise 5.1.7

Describe the congruence classes in \(F[x]\) modulo the polynomial \(x\).

Exercise 5.1.8

Prove or disprove: If \(p(x)\) is relatively prime to \(k(x)\) and \(f(x)k(x) \equiv g(x)k(x) \pmod{p(x)}\), then \(f(x) \equiv g(x) \pmod{p(x)}\).

Exercise 5.1.9

Prove that \(f(x) \equiv g(x) \pmod{p(x)}\) if and only if \(f(x)\) and \(g(x)\) leave the same remainder when divided by \(p(x)\).

Exercise 5.1.10

Prove or disprove: If \(p(x)\) is irreducible in \(F[x]\) and \(f(x)g(x) \equiv 0_F \pmod{p(x)}\), then \(f(x) \equiv 0_F \pmod{p(x)}\) or \(g(x) \equiv 0_F \pmod{p(x)}\).

Exercise 5.1.11

If \(p(x)\) is reducible in \(F[x]\), prove that there exist \(f(x), g(x) \in F[x]\) such that \(f(x) \not\equiv 0_F \pmod{p(x)}\) and \(g(x) \not\equiv 0_F \pmod{p(x)}\) but \(f(x)g(x) \equiv 0_F \pmod{p(x)}\).

Exercise 5.1.12

If \(f(x)\) is relatively prime to \(p(x)\), prove that there is a polynomial \(g(x) \in F[x]\) such that \(f(x)g(x) \equiv 1_F \pmod{p(x)}\).

Exercise 5.1.13

Suppose \(f(x), g(x) \in \mathbb{R}[x]\) and \(f(x) \equiv g(x) \pmod{p(x)}\). What can be said about the graphs of \(y = f(x)\) and \(y = g(x)\)?