ยง4.1 Exercises

Exercises 4.1.A

Exercise 4.1.1

Perform the indicated operation and simplify your answer:

  1. (a) \((3x^4 + 2x^3 - 4x^2 + x + 4) + (4x^3 + x^2 + 4x + 3)\) in \(\mathbb{Z}_5[x]\).
  2. (b) \((x + 1)^3\) in \(\mathbb{Z}_5[x]\).
  3. (c) \((x - 1)^5$ in $\mathbb{Z}_5[x]\).
  4. (d) \((x^2 - 3x + 2)(2x^3 - 4x + 1)$ in $\mathbb{Z}_7[x]\).

Exercise 4.1.2

Show that the set of all real numbers of the form

\[ a_0 + a_1\pi + a_2\pi^2 + \dots + a_n\pi^n, \]

with \(n \geq 0\) and \(a_i \in \mathbb{Z}\), is a subring of \(R\) that contains both $\mathbb{Z}$ and \(pi\).

Exercise 4.1.3

  • (a) List all polynomials of degree 3 in \(\mathbb{Z}_3[x]\).
  • (b) List all polynomials of degree less than 3 in \(\mathbb{Z}_3[x]\).

Exercise 4.1.4

In each part, give an example of polynomials \(f(x)\), \(g(x) \in \mathbb{Q}[x]\) that satisfy the given condition:

  1. The degree of \(f(x) + g(x)\) is less than the maximum of \(\text{deg}(f(x))\) and \(\text{deg}(g(x))\).
  2. \(\text{Deg}(f(x) + g(x)) = \max(\text{deg}(f(x)), \text{deg}(g(x)))\).
  3. \(\text{Deg}(f(x)g(x)) = \text{deg}(f(x)) + \text{deg}(g(x))\).
  4. Find polynomials \(f(x)\) and \(g(x)\) such that \(f(x)g(x) = g(x)q(x) + r(x)\), and \(r(x) = 0\) or \(\text{deg}(r(x)) < \text{deg}(g(x))\).

Exercise 4.1.5

Find polynomials \(f(x)\) and \(g(x)\) such that \(f(x)g(x) = g(x)q(x) + r(x)\), and \(r(x) = 0\) or \(\text{deg}(r(x)) < \text{deg}(g(x))\):

  1. \(f(x) = 3x^4 - 2x^3 + 6x^2 - x + 2\) and \(g(x) = x^2 + x + 1\) in \(\mathbb{Z}_3[x]\).
  2. \(f(x) = x^4 - 7x + 1\) and \(g(x) = 2x^2 + 1\) in \(\mathbb{Z}_5[x]\).
  3. \(f(x) = 2x^4 + x^3 + x^2 + 2x - 1\) and \(g(x) = 2x - 1\) in \(\mathbb{Z}_7[x]\).
  4. \(f(x) = 4x^4 + 2x^3 + 6x^2 + 4x + 5\) and \(g(x) = 3x^2 + 2\) in \(\mathbb{Z}_2[x]\).

Exercise 4.1.6

Which of the following subsets of \(R[x]\) are subrings of \(R[x]\)? Justify your answer:

  1. All polynomials with constant term \(0_R\).
  2. All polynomials of degree 2.
  3. All polynomials of degree \(\leq k\), where \(k\) is a fixed positive integer.
  4. All polynomials in which the odd powers of \(x\) have zero coefficients.
  5. All polynomials in which the even powers of \(x\) have zero coefficients.

Exercise 4.1.7

Let \(R\) be commutative. Show that \(R[x]\) is also commutative.

Exercise 4.1.8

If \(R\) has multiplicative identity \(1_R\), show that \(1_R\) is also the multiplicative identity of \(R[x]\).

Exercise 4.1.9

If \(c \in R\) is a zero divisor in a commutative ring \(R\), then is \(c\) also a zero divisor in \(R[x]\)?

Exercise 4.1.10

If \(F\) is a field, show that \(F[x]\) is not a field. [Hint: Is \(x\) a unit in \(F[x]\)?]

Exercises 4.1.B

Exercise 4.1.11

Show that \(1 + 3x\) is a unit in \(\mathbb{Z}_6[x]\). Hence, Corollary 4.5 may be false if \(R\) is not an integral domain.

Exercise 4.1.12

If \(f(x), g(x) \in R[x]\) and \(f(x) + g(x) \neq 0_R\), show that

\[ \text{deg}(f(x) + g(x)) \leq \max(\text{deg}(f(x)), \text{deg}(g(x)). \]

Exercise 4.1.13

Let \(R\) be a commutative ring. If \(a_n \neq 0_R\) and

\[ f(x)=a_0 + a_1x + a_2x^2 + \dots + a_nx^n, \]

with \(a_n \neq 0_R\) is a zero divisor in \(R[x]\), prove that \(a_n\) is a zero divisor in \(R\).

Exercise 4.1.14

Answer the following questions:

  1. (a) Let \(R\) be an integral domain and \(f(x), g(x) \in R[x]\). Assume that the leading coefficient of \(g(x)\) is a unit in \(R\). Verify that the Division Algorithm holds for \(f(x)\) as dividend and \(g(x)\) as divisor. [Hint: Adapt the proof of Theorem 4.6. Where is the hypothesis that \(F\) is a field used there?]
  2. (b) Give an example in \(\mathbb{Z}[x]\) to show that part (a) may be false if the leading coefficient of \(g(x)\) is not a unit. [Hint: Exercise 5(b) with \(\mathbb{Z}\) in place of \(\mathbb{Q}\).]

Exercise 4.1.15

Let \(R\) be a commutative ring with identity and \(a \in R\). (a) If \(a^3=0_R\), show that \(1_R + ax\) is a unit in \(R[x]\). [Hint: Consider \(1 - ax + a^2x^2\).] (b) If \(a^4=0_R\), show that \(1_R + ax\) is a unit in \(R[x]\).

Exercise 4.1.16

Let \(R\) be a commutative ring with identity and \(a \in R\). If \(1_R + ax\) is a unit in \(R[x]\), show that \(a^n=0_R\) for some integer \(n> 0\). [Hint: Suppose that the inverse of \(1_R + ax\) is \(b_0 + b_1x + b_2x^2 + \dots + b_kx^k\). Since their product is \(1_R\), \(b_0 = 1_R\) (Why?) and the other coefficients are all \(0_R\].

Exercise 4.1.17

Let \(R\) be an integral domain. Assume that the Division Algorithm always holds in \(R[x]\). Prove that \(R\) is a field.

Exercise 4.1.18

Let \(\varphi: R[x] \to R\) be the function that maps each polynomial in \(R[x]\) onto its constant term (an element of \(R\)). Show that \(\varphi\) is a surjective homomorphism of rings.

Exercise 4.1.19

Let \(\varphi: \mathbb{Z}[x] \to \mathbb{Z}_n[x]\) be the function that maps the polynomial \(a_0 + a_1x + \dots + a_kx^k\) in \(\mathbb{Z}[x]\) onto the polynomial \([a_0] + [a_1]x + \dots + [a_k]x^k\), where \([a_i]\) denotes the class of the integer \(a_i\) in \(\mathbb{Z}_n\). Show that \(\varphi\) is a surjective homomorphism of rings.

Exercise 4.1.20

Let \(D: \mathbb{R}[x] \to \mathbb{R}[x]\) be the derivative map defined by

\[ D(a_0 + a_1x + a_2x^2 + \dots + a_nx^n) = a_1 + 2a_2x + 3a_3x^2 + \dots + na_nx^{n-1}. \]

Is \(D\) a homomorphism of rings? An isomorphism?

Exercises 4.1.C

Exercise 4.1.21

Let \(h: R \to S\) be a homomorphism of rings and define a function \(\bar{h}: R[x] \to S[x]\) by the rule

\[ \bar{h}(a_0 + a_1x + \dots + a_nx^n) = h(a_0) + h(a_1)x + \dots + h(a_n)x^n. \]

Prove that:

  1. \(\bar{h}\) is a homomorphism of rings.
  2. \(\bar{h}\) is injective if and only if \(h\) is injective.
  3. \(\bar{h}\) is surjective if and only if \(h\) is surjective.
  4. If \(R \cong S\), then \(R[x] \cong S[x]\).

Exercise 4.1.22

Let \(R\) be a commutative ring and let \(k(x)\) be a fixed polynomial in \(R[x]\). Prove that there exists a unique homomorphism \(\varphi: R[x] \to R[x]\) such that

\[ \varphi(r) = r \text{ for all } r \in R \quad \text{and} \quad \varphi(x) = k(x). \]