22.4: Exercises

1

Calculate each of the following.

1. \(\displaystyle [\mathrm{GF}(3^6) : \mathrm{GF}(3^3)]\)
2. \(\displaystyle [\mathrm{GF}(128): \mathrm{GF}(16)]\)
3. \(\displaystyle [\mathrm{GF}(625) : \mathrm{GF}(25) ]\)
4. \(\displaystyle [\mathrm{GF}(p^{12}): \mathrm{GF}(p^2)]\)

2

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

3

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

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 \({\mathbb Z}_2(\alpha)\text{.}\)

5

Construct a finite field of order \(27\text{.}\)

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

8

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

9

Determine the number of cyclic codes of length \(n\) for \(n = 6, 7, 8, 10\text{.}\)

10

Prove that the ideal \(\langle t + 1 \rangle\) in \(R_n\) is the code in \({\mathbb Z}_2^n\) consisting of all words of even parity.

11

Construct all BCH codes of

1. length \(7\text{.}\)
2. length \(15\text{.}\)

12

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

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 \(E\text{.}\)

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.

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 \(\Phi : \mathrm{GF}(p^n) \rightarrow \mathrm{GF}(p^n)\) given by \(\Phi : \alpha \mapsto \alpha^p\) is an automorphism of order \(n\text{.}\)

22

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

23

Let \(E\) and \(F\) be subfields of \(\mathrm{GF}(p^n)\text{.}\) 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 \((p-1)! \equiv -1 \pmod{p}\text{.}\)

25

If \(g(t)\) is the minimal generator polynomial for a cyclic code \(C\) in \(R_n\text{,}\) prove that the constant term of \(g(x)\) is \(1\text{.}\)

26

Often it is conceivable that a burst of errors might occur during transmission, as in the case of a power surge. Such a momentary burst of interference might alter several consecutive bits in a codeword. Cyclic codes permit the detection of such error bursts. Let \(C\) be an \((n,k)\)-cyclic code. Prove that any error burst up to \(n-k\) digits can be detected.

27

Prove that the rings \(R_n\) and \({\mathbb Z}_2^n\) are isomorphic as vector spaces.

28

Let \(C\) be a code in \(R_n\) that is generated by \(g(t)\text{.}\) If \(\langle f(t) \rangle\) is another code in \(R_n\text{,}\) show that \(\langle g(t) \rangle \subset \langle f(t) \rangle\) if and only if \(f(x)\) divides \(g(x)\) in \({\mathbb Z}_2[x]\text{.}\)

29

Let \(C = \langle g(t) \rangle\) be a cyclic code in \(R_n\) and suppose that \(x^n - 1 = g(x) h(x)\text{,}\) where \(g(x) = g_0 + g_1 x + \cdots + g_{n - k} x^{n - k}\) and \(h(x) = h_0 + h_1 x + \cdots + h_k x^k\text{.}\) Define \(G\) to be the \(n \times k\) matrix

\[ G = \begin{pmatrix} g_0 & 0 & \cdots & 0 \\ g_1 & g_0 & \cdots & 0 \\ \vdots & \vdots &\ddots & \vdots \\ g_{n-k} & g_{n-k-1} & \cdots & g_0 \\ 0 & g_{n-k} & \cdots & g_{1} \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & g_{n-k} \end{pmatrix} \nonumber \]

and \(H\) to be the \((n-k) \times n\) matrix

\[ H = \begin{pmatrix} 0 & \cdots & 0 & 0 & h_k & \cdots & h_0 \\ 0 & \cdots & 0 & h_k & \cdots & h_0 & 0 \\ \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ h_k & \cdots & h_0 & 0 & 0 & \cdots & 0 \end{pmatrix}\text{.} \nonumber \]

1. Prove that \(G\) is a generator matrix for \(C\text{.}\)
2. Prove that \(H\) is a parity-check matrix for \(C\text{.}\)
3. Show that \(HG = 0\text{.}\)