ยง4.4 Exercises
Note: \(F\) denotes a field.
Exercises 4.3.A
Exercise 4.4.1
(a) Find a nonzero polynomial in \(\mathbb{Z}_2[x]\) that induces the zero function on \(\mathbb{Z}_2\).
(b) Do the same in \(\mathbb{Z}_3[x]\).
Exercise 4.4.2
Find the remainder when \(f(x)\) is divided by \(g(x)\):
(a) \(f(x) = x^{10} + x^8\) and \(g(x) = x - 1\) in \(\mathbb{Q}[x]\)
(b) \(f(x) = 2x^5 - 3x^4 + x^3 - 8x + 2\) and \(g(x) = x - 8\) in \(\mathbb{Q}[x]\)
(c) \(f(x) = 10x^5 - 8x^5 + 6x^4 + 4x^{37} - 2x^{15} + 5\) and \(g(x) = x + 1\) in \(\mathbb{Q}[x]\)
(d) \(f(x) = 2x^4 + 3x^2 + 2x + 3\) and \(g(x) = x - 3\) in \(\mathbb{Z}_5[x]\)
Exercise 4.4.3
Determine if \(h(x)\) is a factor of \(f(x)\):
(a) \(h(x) = x + 2\) and \(f(x) = x^3 - 3x^2 - 4x - 12\) in \(\mathbb{R}[x]\)
(b) \(h(x) = x - \frac{1}{2}\) and \(f(x) = 2x^4 + x^3 + \frac{3}{4}x - \frac{3}{2}\) in \(\mathbb{Q}[x]\)
(c) \(h(x) = x + 2\) and \(f(x) = 3x^5 + 4x^4 + 2x^3 - x^2 + 2x + 1\) in \(\mathbb{Z}_5[x]\)
(d) \(h(x) = x - 3\) and \(f(x) = x^6 - 3x^4 - x^3 - 5\) in \(\mathbb{Z}_7[x]\)
Exercise 4.4.4
For what value of \(k\) is \(x - 2\) a factor of \(x^4 + 3x^3 - 5x^2 + 3x + k\) in \(\mathbb{Z}_7[x]\)?
Exercise 4.4.5
For what value of \(k\) is \(x + 1\) a factor of \(x^4 + 2x^3 - 3x^2 + kx + 1\) in \(\mathbb{Z}_5[x]\)?
Exercise 4.4.6
Show that \(x - 1\) divides \(a_0x^n + a_1x^{n-1} + a_2x^2 + \cdots + a_n\) in \(F[x]\) if and only if \(a_0 + a_1 + a_2 + \cdots + a_n = 0_F\).
Exercise 4.4.7
Use the Factor Theorem to show that \(x^7 - x\) factors in \(\mathbb{Z}_7[x]\) as \((x - 0)(x - 2)(x - 3)(x - 4)(x - 5)(x - 6)\), without doing any polynomial multiplication.
Exercise 4.4.8
Use the Factor Theorem to show that \(x^7 - x\) factors in \(\mathbb{Z}_7[x]\) as \((x - 0)(x - 1)(x - 2)(x - 3)(x - 4)(x - 5)(x - 6)\), without doing any polynomial multiplication.
Exercise 4.4.9
Use the Factor Theorem to show that \(x^7 - x\) factors in \(\mathbb{Z}_7[x]\) as \((x - 0)(x - 1)(x - 2)(x - 3)(x - 4)(x - 5)(x - 6)\), without doing any polynomial multiplication.
Exercise 4.4.10
Use the Factor Theorem to show that \(x^7 - x\) factors in \(\mathbb{Z}_7[x]\) as \((x - 0)(x - 1)(x - 2)(x - 3)(x - 4)(x - 5)(x - 6)\), without doing any polynomial multiplication.
Exercise 4.4.11
Use the Factor Theorem to show that \(x^7 - x\) factors in \(\mathbb{Z}_7[x]\) as \((x - 0)(x - 1)(x - 2)(x - 3)(x - 4)(x - 5)(x - 6)\), without doing any polynomial multiplication.
Exercises 4.3.B
Exercise 4.4.12
If \(a \in F\) is a nonzero root of \(c_nx^n + c_{n-1}x^{n-1} + \cdots + c_1x + c_0\), prove that \(a^{-1}\) is a root of \(c_0x^n + c_1x^{n-1} + \cdots + c_{n-1}x + c_n\).
Exercise 4.4.13
(a) If \(f(x)\) and \(g(x)\) are associates in \(F[x]\), show that they have the same roots in \(F\).
(b) If \(f(x), g(x) \in F[x]\) have the same roots in \(F\), are they associates in \(F[x]\)?
Exercise 4.4.14
(a) Suppose \(r, s \in F\) are roots of \(ax^2 + bx + c \in F[x]\) (with \(a \neq 0_F\)). Use the Factor Theorem to show that \(r + s = -a^{-1}b\) and \(rs = a^{-1}c\).
Exercise 4.4.15
Prove that \(x^2 + 1\) is reducible in \(\mathbb{Z}_n[x]\) if and only if there exist integers \(a\) and \(b\) such that \(p = a^2 + b^2\) and \(a \equiv b \pmod{p}\).
Exercise 4.4.16
Let \(f(x), g(x) \in F[x]\) have degree \(\leq n\) and let \(c_1, \ldots, c_n\) be distinct elements of \(F\). If \(f(c_i) = g(c_i)\) for \(i = 0, 1, \ldots, n\), prove that \(f(x) = g(x)\) in \(F[x]\).
Exercise 4.4.17
Find a polynomial of degree 2 in \(\mathbb{Z}_6[x]\) that has four roots in \(\mathbb{Z}_6\). Does this contradict Corollary 4.17?
Exercise 4.4.18
Let \(\varphi: \mathbb{C} \rightarrow \mathbb{C}\) be an isomorphism of rings such that \(\varphi(a) = a\) for each \(a \in \mathbb{Q}\). Suppose \(r \in \mathbb{C}\) is a root of \(f(x) \in \mathbb{Q}[x]\). Prove that \(\varphi(r)\) is also a root of \(f(x)\).
Exercise 4.4.19
We say that \(a \in F\) is a multiple root of \(f(x) \in F[x]\) if \((x - a)^k\) is a factor of \(f(x)\) for some \(k \geq 2\).
(a) Prove that \(a \in F\) is a multiple root of \(f(x) \in R[x]\) if and only if \(a\) is a root of both \(f(x)\) and \(f'(x)\), where \(f'(x)\) is the derivative of \(f(x)\).
(b) If \(f(x) \in F[x]\) and if \(f(x)\) is relatively prime to \(f'(x)\), prove that \(f(x)\) has no multiple root in \(R\).
Exercise 4.4.20
Let \(R\) be an integral domain. Then the Division Algorithm holds in \(R[x]\) whenever the divisor is monic, by Exercise 14 in Section 4.1. Use this fact to show that the Remainder and Factor Theorems hold in \(R[x]\).
Exercise 4.4.21
If \(R\) is an integral domain and if \(f(x)\) is a nonzero polynomial of degree \(n\) in \(R[x]\), prove that \(f(x)\) has at most \(n\) roots in \(R\).
Hint: Exercise 20
Exercise 4.4.22
Show that Corollary 4.20 holds if \(F\) is an infinite integral domain.
Hint: See Exercise 21.
Exercise 4.4.23
Let \(f(x), g(x), h(x) \in F[x]\) and \(r \in F\).
(a) If \(f(x) = g(x) + h(x)\) in \(F[x]\), show that \(f(r) = g(r) + h(r)\) in \(F\).
(b) If \(f(x) = g(x)h(x)\) in \(F[x]\), show that \(f(r) = g(r)h(r)\) in \(F\).
Where were these facts used in this section?
Exercise 4.4.24
Let \(a\) be a fixed element of \(F\) and define a map \(\varphi_a: F[x] \rightarrow F\) by \(\varphi_a(f(x)) = f(a)\). Prove that \(\varphi_a\) is a surjective homomorphism of rings. The map \(\varphi_a\) is called an evaluation homomorphism; there is one for each \(a \in F\).
Exercise 4.4.25
Let \(\mathbb{Q}[\pi]\) be the set of all real numbers of the form
\(r_0 + r_1\pi + r_2\pi^2 + \cdots + a_n\pi^n\),
with \(n \geq 0\) and \(r_i \in \mathbb{Q}\).
(a) Show that \(\mathbb{Q}[\pi]\) is a subring of \(\mathbb{R}\).
(b) Show that the function \(\theta: \mathbb{Q}[\pi] \rightarrow \mathbb{Q}[\pi]\) defined by \(\theta(f(x)) = f(\pi)\) is an isomorphism. You may assume the following nontrivial fact: \(\pi\) is not the root of any nonzero polynomial with rational coefficients. Therefore, Theorem 4.1 is true with \(R = \mathbb{Q}\) and \(\pi\) in place of \(x\). However, see Exercise 26.
Exercise 4.4.26
Let \(\mathbb{Q}[\sqrt{2}]\) be the set of all real numbers of the form
\(r_0 + r_1\sqrt{2} + r_2(\sqrt{2})^2 + \cdots + r_n(\sqrt{2})^n\),
with \(n \geq 0\) and \(r_i \in \mathbb{Q}\).
(a) Show that \(\mathbb{Q}[\sqrt{2}]\) is a subring of \(\mathbb{R}\).
(b) Show that the function \(\theta: \mathbb{Q}[x] \rightarrow \mathbb{Q}[\sqrt{2}]\) defined by \(\theta(f(x)) = f(\sqrt{2})\) is a surjective homomorphism, but not an isomorphism. Thus Theorem 4.1 is true with \(R = \mathbb{Q}\) and \(\sqrt{2}\) in place of \(x\). Compare this with Exercise 25.
Exercise 4.4.27
Let \(T\) be the set of all polynomial functions from \(F\) to \(F\). Show that \(T\) is a commutative ring with identity, with operations defined as in calculus: For each \(r \in F\),
\((f + g)(r) = f(r) + g(r) \quad \text{and} \quad (fg)(r) = f(r)g(r)\).
Exercise 4.4.28
Let \(T\) be the ring of all polynomial functions from \(\mathbb{Z}_3\) to \(\mathbb{Z}_3\) (see Exercise 27).
(a) Show that \(T\) is a finite ring with zero divisors. Hint: Consider \(f(x) = x + 1\) and \(g(x) = x^2 + 2x\).
(b) Show that \(T\) cannot possibly be isomorphic to \(\mathbb{Z}_3[x]\). Then see Exercise 30.
Exercise 4.4.29
Use mathematical induction to prove Corollary 4.17.
Exercises 4.3.C
Exercise 4.4.30
If \(F\) is an infinite field, prove that the polynomial ring \(F[x]\) is isomorphic to the ring \(T\) of all polynomial functions from \(F\) to \(F\) (Exercise 27). Hint: Define a map \(\varphi: F[x] \rightarrow T\) by assigning to each polynomial \(f(x) \in F[x]\) its induced function in \(T\); \(\varphi\) is injective by Corollary 4.20.
Exercise 4.4.31
Let \(\varphi: F[x] \rightarrow F[x]\) be an isomorphism such that \(\varphi(a) = a\) for every \(a \in F\). Prove that \(f(x)\) is irreducible in \(F[x]\) if and only if \(\varphi(f(x))\) is.
Exercise 4.4.32
(a) Show that the map \(\varphi: F[x] \rightarrow F[x]\) given by \(\varphi(f(x)) = f(x + 1_F)\) is an isomorphism such that \(\varphi(a) = a\) for every \(a \in F\).
(b) Use Exercise 31 to show that \(f(x)\) is irreducible in \(F[x]\) if and only if \(f(x + 1_F)\) is.