§4.5 Exercises

Exercise 4.5.1

Use the Rational Root Test to write each polynomial as a product of irreducible polynomials in \(\mathbb{Q}[x]\):

(a) \(x^4 - x^3 + x^2 + x + 2\)
(b) \(x^5 + 4x^4 + x^3 - x^2\)
(c) \(3x^5 + 2x^4 - 7x^3 + 2x^2\)
(d) \(2x^4 - 5x^3 + 3x^2 + 4x - 6\)
(e) \(2x^4 + 7x^3 + 5x^2 + 7x + 3\)
(f) \(6x^6 - 31x^4 + 25x^2 + 33x + 7\)

Exercise 4.5.2

Show that \(\sqrt{p}\) is irrational for every positive prime integer \(p\). [Hint: What are the roots of \(x^2 - p\)? Do you prefer this proof to the one in Exercises 30 and 31 of Section 1.3?]

Exercise 4.5.3

If a monic polynomial with integer coefficients has a root in \(\mathbb{Q}\), show that this root must be an integer.

Exercise 4.5.4

Show that each polynomial is irreducible in \(\mathbb{Q}[x]\), as in Example 3:

(a) \(x^4 + 2x^3 + x + 1\)
(b) \(x^4 - 2x^2 + 8x + 1\)

Exercise 4.5.5

Use Eisenstein's Criterion to show that each polynomial is irreducible in \(\mathbb{Q}[x]\):

(a) \(x^5 - 4x + 22\)
(b) \(10 - 15x + 25x^2 - 7x^4\)
(c) \(5x^{11} - 6x^4 + 12x^3 + 36x - 6\)

Exercise 4.5.6

Show that there are infinitely many integers \(k\) such that \(x^9 + 12x^5 - 21x + k\) is irreducible in \(\mathbb{Q}[x]\).

Exercise 4.5.7

Show that each polynomial \(f(x)\) is irreducible in \(\mathbb{Q}[x]\) by finding a prime \(p\) such that \(f(x)\) is irreducible in \(\mathbb{Z}_p[x]\):

(a) \(x^2 + 6x^4 - 4x + 0\)
(b) \(9x^4 + 4x\)

Exercise 4.5.8

Give an example of a polynomial \(f(x) \in \mathbb{Z}[x]\) and a prime \(p\) such that \(f(x)\) is reducible in \(\mathbb{Q}[x]\) but \(\bar{f}(x)\) is irreducible in \(\mathbb{Z}_p[x]\). Does this contradict Theorem 4.25?

Exercise 4.5.9

Give an example of a polynomial in \(\mathbb{Z}[x]\) that is irreducible in \(\mathbb{Q}[x]\) but factors when reduced mod 2, 3, 4, and 5.

Exercise 4.5.10

If a monic polynomial with integer coefficients factors in \(\mathbb{Z}[x]\) as a product of polynomials of degrees \(m\) and \(n\), prove that it can be factored as a product of monic polynomials of degrees \(m\) and \(n\) in \(\mathbb{Z}[x]\).

Exercise 4.5.11

Prove that \(30x^n - 91\) (where \(n \in \mathbb{Z}, n > 1\)) has no roots in \(\mathbb{Q}\).

Exercise 4.5.12

Let \(F\) be a field and \(f(x) \in F[x]\). If \(c \in F\) and \(f(x + c)\) is irreducible in \(F[x]\), prove that \(f(x)\) is irreducible in \(F[x]\). [Hint: Prove the contrapositive.]

Exercise 4.5.13

Prove that \(f(x) = x^4 + 4x + 1\) is irreducible in \(\mathbb{Q}[x]\) by using Eisenstein's Criterion to show that \(f(x + 1)\) is irreducible and applying Exercise 12.

Exercise 4.5.14

Prove that \(f(x) = x^4 + x^3 + x^2 + x + 1\) is irreducible in \(\mathbb{Q}[x]\). [Hint: Use the hint for Exercise 21 with \(p = 5.\]

Exercise 4.5.15

Let \(f(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0\) be a polynomial with integer coefficients. If \(p\) is a prime such that \(p | a_1, p | a_2, \dots, p | a_n\), but \(p \nmid a_0\) and \(p^2 \nmid a_0\), prove that \(f(x)\) is irreducible in \(\mathbb{Q}[x]\).

Exercise 4.5.16

Show by example that this statement is false: If \(f(x) \in \mathbb{Z}[x]\) and there is no prime \(p\) satisfying the hypotheses of Theorem 4.24, then \(f(x)\) is reducible in \(\mathbb{Q}[x]\).

Exercise 4.5.17

Show that there are \(n^{n+1} - n^k\) polynomials of degree \(k\) in \(\mathbb{Z}_p[x]\).

Exercise 4.5.18

Which of these polynomials are irreducible in \(\mathbb{Q}[x]\):

(a) \(x^4 - x^3 + x^2 + 1\)
(b) \(x^4 + x + 1\)
(c) \(x^5 + 4x^4 + 2x^3 + 3x^2 + 5\)
(d) \(x^5 + 5x^2 + 4x + 7\)

Exercise 4.5.19

Write each polynomial as a product of irreducible polynomials in \(\mathbb{Q}[x]\):

(a) \(x^5 + 2x^4 - 6x^2 - 16x - 8\)
(b) \(x^7 - 2x^6 - 6x^4 - 15x^2 - 33x - 9\)

Exercise 4.5.20

If \(f(x) = a_nx^n + \cdots + a_1x + a_0\), \(g(x) = b_mx^m + \cdots + b_1x + b_0\), and \(h(x) = c_px^p + \cdots + c_1x + c_0\) are polynomials in \(\mathbb{Z}[x]\) such that \(f(x) = g(x)h(x)\), show that in \(\mathbb{Z}[x]\), \(\bar{f}(x) = \bar{g}(x)\bar{h}(x)\). Also, see Exercise 19 in Section 4.1.

Exercise 4.5.21

Prove that for a prime \(p\), \(f(x) = x^p + 1 + x^{p-2} + \cdots + x^2 + x + 1\) is irreducible in \(\mathbb{Q}[x]\). [Hint: Let \(f(x + 1) = x^p + \cdots + (p-1)!\) and show that \(f(x + 1)\) is irreducible. Expand \((x + 1)^p\) by the Binomial Theorem (Appendix E) and note that \(p\) divides \(\binom{p}{k}\) when \(k > 0\). Use Eisenstein's Criterion to show that \(f(x + 1)\) is irreducible; apply Exercise 12.]