Skip to main content

Exercises 5.5 Exercises

1.

List all of the polynomials of degree \(3\) or less in \({\mathbb Z}_2[x]\text{.}\)

2.

Compute each of the following.

  1. \((5x^2 + 3x - 4) + (4x^2 - x + 9)\) in \({\mathbb Z}_{12}\)

  2. \((5x^2 + 3x - 4) (4x^2 - x + 9)\) in \({\mathbb Z}_{12}\)

  3. \((7x^3 + 3x^2 - x) + (6x^2 - 8x + 4)\) in \({\mathbb Z}_9\)

  4. \((3x^2 + 2x - 4) + (4x^2 + 2)\) in \({\mathbb Z}_5\)

  5. \((3x^2 + 2x - 4) (4x^2 + 2)\) in \({\mathbb Z}_5\)

  6. \((5x^2 + 3x - 2)^2\) in \({\mathbb Z}_{12}\)

3.

Use the division algorithm to find \(q(x)\) and \(r(x)\) such that \(a(x) = q(x) b(x) + r(x)\) with \(\deg r(x) \lt \deg b(x)\) for each of the following pairs of polynomials.

  1. \(a(x) = 5 x^3 + 6x^2 - 3 x + 4\) and \(b(x) = x - 2\) in \({\mathbb Z}_7[x]\)

  2. \(a(x) = 6 x^4 - 2 x^3 + x^2 - 3 x + 1\) and \(b(x) = x^2 + x - 2\) in \({\mathbb Z}_7[x]\)

  3. \(a(x) = 4 x^5 - x^3 + x^2 + 4\) and \(b(x) = x^3 - 2\) in \({\mathbb Z}_5[x]\)

  4. \(a(x) = x^5 + x^3 -x^2 - x\) and \(b(x) = x^3 + x\) in \({\mathbb Z}_2[x]\)

4.

Find the greatest common divisor of each of the following pairs \(p(x)\) and \(q(x)\) of polynomials.

  1. \(p(x) = x^3 - 6x^2 + 14x - 15\) and \(q(x) = x^3 - 8x^2 + 21x - 18\text{,}\) where \(p(x), q(x) \in {\mathbb Q}[x]\)

  2. \(p(x) = x^3 + x^2 - x + 1\) and \(q(x) = x^3 + x - 1\text{,}\) where \(p(x), q(x) \in {\mathbb Z}_2[x]\)

  3. \(p(x) = x^3 + x^2 - 4x + 4\) and \(q(x) = x^3 + 3 x -2\text{,}\) where \(p(x), q(x) \in {\mathbb Z}_5[x]\)

  4. \(p(x) = x^3 - 2 x + 4\) and \(q(x) = 4 x^3 + x + 3\text{,}\) where \(p(x), q(x) \in {\mathbb Q}[x]\)

5.

Find all of the zeros for each of the following polynomials.

  1. \(5x^3 + 4x^2 - x + 9\) in \({\mathbb Z}_{12}\)

  2. \(3x^3 - 4x^2 - x + 4\) in \({\mathbb Z}_{5}\)

  3. \(5x^4 + 2x^2 - 3\) in \({\mathbb Z}_{7}\)

  4. \(x^3 + x + 1\) in \({\mathbb Z}_2\)

6.

Find all of the units in \({\mathbb Z}[x]\text{.}\)

7.

Find a unit \(p(x)\) in \({\mathbb Z}_4[x]\) such that \(\deg p(x) \gt 1\text{.}\)

8.

Find all of the irreducible polynomials of degrees \(2\) and \(3\) in \({\mathbb Z}_2[x]\text{.}\)

9.

Give two different factorizations of \(x^2 + x + 8\) in \({\mathbb Z}_{10}[x]\text{.}\)

10.

Prove or disprove: There exists a polynomial \(p(x)\) in \({\mathbb Z}_6[x]\) of degree \(n\) with more than \(n\) distinct zeros.

11.

Let \(p\) be prime and denote the field of fractions of \({\mathbb Z}_p[x]\) by \({\mathbb Z}_p(x)\text{.}\) Prove that \({\mathbb Z}_p(x)\) is an infinite field of characteristic \(p\text{.}\)

12.

Prove or disprove: \(x^p + a\) is irreducible for any \(a \in {\mathbb Z}_p\text{,}\) where \(p\) is prime.

13.

Let \(F\) be a field and \(a \in F\text{.}\) If \(p(x) \in F[x]\text{,}\) show that \(p(a)\) is the remainder obtained when \(p(x)\) is divided by \(x - a\text{.}\)

14.

If \(F\) is a field, show that there are infinitely many irreducible polynomials in \(F[x]\text{.}\)

15.

Which of the following polynomials are irreducible over \({\mathbb Q}[x]\text{?}\)

  1. \(\displaystyle x^4 - 2x^3 + 2x^2 + x + 4\)

  2. \(\displaystyle x^4 - 5x^3 + 3x - 2\)

  3. \(\displaystyle 3x^5 - 4x^3 - 6x^2 + 6\)

  4. \(\displaystyle 5x^5 - 6x^4 - 3x^2 + 9 x - 15\)

16.

If \(F\) is a field, show that \(F[x_1, \ldots, x_n]\) is an integral domain.

17.

Show that the division algorithm does not hold for \({\mathbb Z}[x]\text{.}\) Why does it fail?

18.

Let \(R\) be a commutative ring. Prove that multiplication is commutative in \(R[x]\text{.}\)

19.

Let \(R\) be a commutative ring. Prove that multiplication is distributive in \(R[x]\text{.}\)

20.

Show that \(x^p - x\) has \(p\) distinct zeros in \({\mathbb Z}_p\text{,}\) for any prime \(p\text{.}\) Conclude that

\begin{equation*} x^p - x = x(x - 1)(x - 2) \cdots (x - (p - 1)). \end{equation*}

21.

Let \(F\) be a field and \(f(x) = a_0 + a_1 x + \cdots + a_n x^n\) be in \(F[x]\text{.}\) Define \(f'(x) = a_1 + 2 a_2 x + \cdots + n a_n x^{n - 1}\) to be the derivative of \(f(x)\text{,}\) where \(2 = 1+1\text{,}\) \(3=1+1+1\) and so on.

  1. Prove that

    \begin{equation*} (f + g)'(x) = f'(x) + g'(x). \end{equation*}

  2. For which polynomials \(f\) does \(f' = 0\) hold if \(\chr F = 0\text{?}\)

  3. For which polynomials \(f\) does \(f' = 0\) hold if \(\chr F = p\text{?}\)

  4. Prove that

    \begin{equation*} (fg)'(x) = f'(x)g(x) + f(x) g'(x). \end{equation*}

  5. Suppose that we can factor a polynomial \(f(x) \in F[x]\) into linear factors, say

    \begin{equation*} f(x) = a(x - a_1) (x - a_2) \cdots ( x - a_n). \end{equation*}

    Prove that \(f(x)\) has no repeated factors if and only if \(f(x)\) and \(f'(x)\) are relatively prime.

22.

Let \(F\) be a field. Show that \(F[x]\) is never a field.

23.

Let \(R\) be an integral domain. Prove that \(R[x_1, \ldots, x_n]\) is an integral domain.

24.

Suppose that \(R\) and \(S\) are isomorphic rings. Prove that \(R[x] \cong S[x]\text{.}\)

25.

Let \(R\) be a commutative ring. Show that \(R[x]\) has a subring \(R'\) isomorphic to \(R\text{.}\)

26.

Let \(p(x)\) and \(q(x)\) be polynomials in \(R[x]\text{,}\) where \(R\) is a commutative ring. Prove that \(\deg( p(x) + q(x) ) \leq \max( \deg p(x), \deg q(x) )\text{.}\)

27.

Determine whether each of the following polynomials is irreducible in \(\mathbb Q[x]\text{:}\)

  1. \(\displaystyle 5 x^5 - 6 x^4 - 3x^2 + 9x - 15\)

  2. \(\displaystyle x^4 - 2x^3 - x^2 + x + 2\)

  3. \(\displaystyle x^4 - 3x^3 + 6x^2 + 5\)

  4. \(\displaystyle x^3 + 7x^2 - 2x + 165\)

  5. \(\displaystyle x^4 - 2 x^2 + 4x + 18\)

  6. \(\displaystyle x^4 + 3x^2 - 9 x + 6\)

  7. \(\displaystyle x^4 + 4 x^3 - 2x^2 + 5 x - 3\)

  8. \(\displaystyle x^5 + 6 x^4 - 12 x^2 + 3 x + 6\)

  9. \(\displaystyle x^4 + 3x^3 + 3x^2 - 3 x + 9\)

28.

From the following list of polynomials, find 4 which are irreducible in \(\mathbb Q[x]\) and explain why they are irreducible:

\begin{align*} f_1 &= x^2 + 3x + 3\\ f_2 &= x^3 - 2 x^2 - x + 4\\ f_3 &= x^3 + 4 x - 2\\ f_4 &= x^3 + 2 x^2 - 4x - 3\\ f_5 &= x^4 - 4x + 3\\ f_6 &= x^4 + 2 x^2 - x - 3 \end{align*}

29.

From the following list of polynomials, find 3 which are irreducible in \(\mathbb Q[x]\) and explain why they are irreducible:

\begin{align*} g_1 &= x^3 + x^2 + 3x + 2\\ g_2 &= x^4 - x^3 - 8 x^2 + 11x - 3\\ g_3 &= x^4 - x^3 - 3x^2 + 5x + 3\\ g_4 &= x^4 + 3x^2 - 9x + 6 \end{align*}

30.

Prove that for every positive integer \(n\text{,}\) there exists an irreducible polynomial of degree \(n\) in \(\mathbb Q[x]\text{.}\)

31. The Rational Root Theorem.

Let

\begin{equation*} p(x) = a_n x^n + a_{n - 1}x^{n - 1} + \cdots + a_0 \in \mathbb Z[x], \end{equation*}

where \(a_n \neq 0\text{.}\) Prove that if \(p(r/s) = 0\text{,}\) where \(\gcd(r, s) = 1\text{,}\) then \(r \mid a_0\) and \(s \mid a_n\text{.}\)

32. Cyclotomic Polynomials.

The polynomial

\begin{equation*} \Phi_n(x) = \frac{x^n - 1}{x - 1} = x^{n - 1} + x^{n - 2} + \cdots + x + 1 \end{equation*}

is called the cyclotomic polynomial. Show that \(\Phi_p(x)\) is irreducible over \({\mathbb Q}\) for any prime \(p\text{.}\)