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Exercises 6.2 Exercises

1.

Calculate each of the following.
  1. \(\displaystyle [\gf(3^6) : \gf(3^3)]\)
  2. \(\displaystyle [\gf(128): \gf(16)]\)
  3. \(\displaystyle [\gf(625) : \gf(25) ]\)
  4. \(\displaystyle [\gf(p^{12}): \gf(p^2)]\)
Hint.
Make sure that you have a field extension.

2.

Calculate \([\gf(p^m): \gf(p^n)]\text{,}\) where \(n \mid m\text{.}\)

3.

What is the lattice of subfields for \(\gf(p^{30})\text{?}\)

4.

Let \(\alpha\) be a zero of \(x^3 + x^2 + 1\) over \({\mathbb Z}_2\text{.}\) Construct a finite field of order \(8\text{.}\) Show that \(x^3 + x^2 + 1\) splits in \({\mathbb Z}_2(\alpha)\text{.}\)
Hint.
There are eight elements in \({\mathbb Z}_2(\alpha)\text{.}\) Exhibit two more zeros of \(x^3 + x^2 + 1\) other than \(\alpha\) in these eight elements.

5.

Construct a finite field of order \(27\text{.}\)
Hint.
Find an irreducible polynomial \(p(x)\) in \({\mathbb Z}_3[x]\) of degree \(3\) and show that \({\mathbb Z}_3[x]/ \langle p(x) \rangle\) has \(27\) elements.

6.

Prove or disprove: \({\mathbb Q}^\ast\) is cyclic.

7.

Factor each of the following polynomials in \({\mathbb Z}_2[x]\text{.}\)
  1. \(\displaystyle x^5- 1\)
  2. \(\displaystyle x^6 + x^5 + x^4 + x^3 + x^2 + x + 1\)
  3. \(\displaystyle x^9 - 1\)
  4. \(\displaystyle x^4 +x^3 + x^2 + x + 1\)
Hint.
(a) \(x^5 -1 = (x+1)(x^4+x^3 + x^2 + x+ 1)\text{;}\) (c) \(x^9 -1 = (x+1)( x^2 + x+ 1)(x^6+x^3+1)\text{.}\)

8.

Prove or disprove: \({\mathbb Z}_2[x] / \langle x^3 + x + 1 \rangle \cong {\mathbb Z}_2[x] / \langle x^3 + x^2 + 1 \rangle\text{.}\)
Hint.
True.

12.

Prove or disprove: There exists a finite field that is algebraically closed.
Hint.
False.

13.

Let \(p\) be prime. Prove that the field of rational functions \({\mathbb Z}_p(x)\) is an infinite field of characteristic \(p\text{.}\)

14.

Let \(D\) be an integral domain of characteristic \(p\text{.}\) Prove that \((a - b)^{p^n} = a^{p^n} - b^{p^n}\) for all \(a, b \in D\text{.}\)

15.

Show that every element in a finite field can be written as the sum of two squares.

16.

Let \(E\) and \(F\) be subfields of a finite field \(K\text{.}\) If \(E\) is isomorphic to \(F\text{,}\) show that \(E = F\text{.}\)

17.

Let \(F \subset E \subset K\) be fields. If \(K\) is a separable extension of \(F\text{,}\) show that \(K\) is also separable extension of \(E\text{.}\)
Hint.
If \(p(x) \in F[x]\text{,}\) then \(p(x) \in E[x]\text{.}\)

18.

Let \(E\) be an extension of a finite field \(F\text{,}\) where \(F\) has \(q\) elements. Let \(\alpha \in E\) be algebraic over \(F\) of degree \(n\text{.}\) Prove that \(F( \alpha )\) has \(q^n\) elements.
Hint.
Since \(\alpha\) is algebraic over \(F\) of degree \(n\text{,}\) we can write any element \(\beta \in F(\alpha)\) uniquely as \(\beta = a_0 + a_1 \alpha + \cdots + a_{n - 1} \alpha^{n - 1}\) with \(a_i \in F\text{.}\) There are \(q^n\) possible \(n\)-tuples \((a_0, a_1, \ldots, a_{n - 1})\text{.}\)

19.

Show that every finite extension of a finite field \(F\) is simple; that is, if \(E\) is a finite extension of a finite field \(F\text{,}\) prove that there exists an \(\alpha \in E\) such that \(E = F( \alpha )\text{.}\)

20.

Show that for every \(n\) there exists an irreducible polynomial of degree \(n\) in \({\mathbb Z}_p[x]\text{.}\)

21.

Prove that the Frobenius map \(\Phi : \gf(p^n) \rightarrow \gf(p^n)\) given by \(\Phi : \alpha \mapsto \alpha^p\) is an automorphism of order \(n\text{.}\)

22.

Show that every element in \(\gf(p^n)\) can be written in the form \(a^p\) for some unique \(a \in \gf(p^n)\text{.}\)

23.

Let \(E\) and \(F\) be subfields of \(\gf(p^n)\text{.}\) If \(|E| = p^r\) and \(|F| = p^s\text{,}\) what is the order of \(E \cap F\text{?}\)

24. Wilson’s Theorem.

Let \(p\) be prime. Prove that \((p-1)! \equiv -1 \pmod{p}\text{.}\)
Hint.
Factor \(x^{p-1} - 1\) over \({\mathbb Z}_p\text{.}\)
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