§4.3 Exercises

Exercise 4.3.1

Find a monic associate of:

1. \((a)\) \(3x^3 + 2x^2 + x + 5\) in \(\mathbb{Q}[x]\)
2. \((b)\) \(3x^5 - 4x^2 + 1\) in \(\mathbb{Z}_5[x]\)
3. \((c)\) \(ix + x - 1\) in \(\mathbb{C}[x]\)

Exercise 4.3.2

Prove that every nonzero \(f(x) \in F[x]\) has a unique monic associate in \(F[x]\).

Exercise 4.3.3

List all associates of:

1. \((a)\) \(x^2 + x + 1\) in \(\mathbb{Z}_5[x]\)
2. \((b)\) \(3x + 2\) in \(\mathbb{Z}_7[x]\)

Exercise 4.3.4

Show that a nonzero polynomial in \(\mathbb{Z}_p[x]\) has exactly \(p - 1\) associates.

Exercise 4.3.5

Prove that \(f(x)\) and \(g(x)\) are associates in \(F[x]\) if and only if \(f(x) \mid g(x)\) and \(g(x) \mid f(x)\).

Exercise 4.3.6

Show that \(x^2 + 1\) is irreducible in \(\mathbb{Q}[x]\). Hint: If not, it must factor as \((ax + b)(cx + d)\) with \(a, b, c, d \in \mathbb{Q}\); show that this is impossible.

Exercise 4.3.7

Prove that \(f(x)\) is irreducible in \(F[x]\) if and only if each of its associates is irreducible.

Exercise 4.3.8

If \(f(x) \in F[x]\) can be written as the product of two polynomials of lower degree, prove that \(f(x)\) is reducible in \(F[x]\). (This is the second part of the proof of Theorem 4.11.)

Exercise 4.3.9

Find all irreducible polynomials of:

1. \((a)\) degree 2 in \(\mathbb{Z}_2[x]\)
2. \((b)\) degree 3 in \(\mathbb{Z}_3[x]\)
3. \((c)\) degree 2 in \(\mathbb{Z}_3[x]\)
4. \((d)\) degree 2 in \(\mathbb{Z}_5[x]\)

Exercise 4.3.10

Is the given polynomial irreducible:

1. \((a)\) \(x^2 - 3\) in \(\mathbb{Q}[x]\)? In \(\mathbb{R}[x]\)? In \(\mathbb{C}[x]\)?
2. \((b)\) \(x^2 + x + 2\) in \(\mathbb{Z}_3[x]\)? In \(\mathbb{Z}_7[x]\)?
3. \((c)\) Show that \(x^3 - 3\) is irreducible in \(\mathbb{Z}_7[x]\).

Exercise 4.3.11

Express \(x^4 - 4\) as a product of irreducibles in \(\mathbb{Q}[x]\), in \(\mathbb{R}[x]\), and in \(\mathbb{C}[x]\).

Exercise 4.3.12

Use unique factorization to find the gcd in \(\mathbb{C}[x]\) of \((x - 3)(x - 4)(x - i)^2\) and \((x - 1)^5 (x - 3)(x - 4)^3\).

Exercise 4.3.13

Show that \(x^2 + x\) can be factored in two ways in \(\mathbb{Z}_2[x]\) as the product of non-constant polynomials that are not units and not associates of \(x\) or \(x + 1\).

Exercise 4.3.14

(a) By counting products of the form \((x + a)(x + b)\), show that there are exactly \((p^2 + p)/2\) monic polynomials of degree 2 that are not irreducible in \(\mathbb{Z}_p[x]\).
(b) Show that there are exactly \((p^2 - p)/2\) monic irreducible polynomials of degree 2 in \(\mathbb{Z}_p[x]\).

Exercise 4.3.15

Prove that \(p(x)\) is irreducible in \(F[x]\) if and only if for every \(g(x) \in F[x]\), either \(p(x) \mid g(x)\) or \(p(x)g(x)\) is relatively prime to \(g(x)\).

Exercise 4.3.16

Prove (1) \(\Rightarrow\) (2) in Theorem 4.12.

Exercise 4.3.17

Without using statement (2), prove directly that statement (1) is equivalent to statement (3) in Theorem 4.12.

Exercise 4.3.18

Prove Corollary 4.13.

Exercise 4.3.19

If \(p(x)\) and \(q(x)\) are nonassociate irreducibles in \(F[x]\), prove that \(p(x)\) and \(q(x)\) are relatively prime.

Exercise 4.3.20

(a) Find a polynomial of positive degree in \(\mathbb{Z}_9[x]\) that is a unit.
(b) Show that every polynomial (except the constant polynomials 3 and 6) in \(\mathbb{Z}_9[x]\) can be written as the product of two polynomials of positive degree.

Exercise 4.3.21

(a) Show that \(x^3 + a\) is reducible in \(\mathbb{Z}_3[x]\) for each \(a \in \mathbb{Z}_3\).
(b) Show that \(x^5 + a\) is reducible in \(\mathbb{Z}_5[x]\) for each \(a \in \mathbb{Z}_5\).

Exercise 4.3.22

Show that \(x^2 + 2\) is irreducible in \(\mathbb{Z}_5[x]\).

Exercise 4.3.23

Factor \(x^4 - 4\) as a product of irreducibles in \(\mathbb{Z}_5[x]\).

Exercise 4.3.24

Prove Theorem 4.14.

Exercise 4.3.25

Prove that every nonconstant \(f(x) \in F[x]\) can be written in the form \(c p_1(x)p_2(x)\cdots p_r(x)\), with \(c \in F\) and each \(p_i(x)\) monic irreducible in \(F[x]\). Show further that if \(f(x) = d q_1(x)q_2(x)\cdots q_m(x)\) with \(d \in F\) and each \(q_i(x)\) monic irreducible in \(F[x]\), then \(m = r\), and after reordering and relabeling if necessary, \(p_i(x)\) is an associate of \(q_i(x)\) for each \(i\).