1
Calculate each of the following.
$[\gf(3^6) : \gf(3^3)]$
$[\gf(128): \gf(16)]$
$[\gf(625) : \gf(25) ]$
$[\gf(p^{12}): \gf(p^2)]$
Hint
Make sure that you have a field extension.
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Section22.3Exercises
¶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. 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.
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 ${\mathbb Z}_2[x]\text{.}$
$x^5- 1$
$x^6 + x^5 + x^4 + x^3 + x^2 + x + 1$
$x^9 - 1$
$x^4 +x^3 + x^2 + x + 1$
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.
9
Determine the number of cyclic codes of length $n$ for $n = 6\text{,}$ 7, 8, 10.
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
length 7.
length 15.
Hint
(a) Use the fact that $x^7 -1 = (x+1)( x^3 + x+ 1)(x^3+x^2+1)\text{.}$
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 $\Phi : \gf(p^n) \rightarrow \gf(p^n)$ 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 $E \cap F\text{?}$
24Wilson's Theorem
Let $p$ be prime. Prove that $(p-1)! \equiv -1 \pmod{p}\text{.}$
Hint
Factor $x^{p-1} - 1$ over ${\mathbb Z}_p\text{.}$
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
\begin{equation*}
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}
\end{equation*}
and $H$ to be the $(n-k) \times n$ matrix
\begin{equation*}
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}.
\end{equation*}
Prove that $G$ is a generator matrix for $C\text{.}$
Prove that $H$ is a parity-check matrix for $C\text{.}$
Show that $HG = 0\text{.}$