Abstract Algebra: Theory and Applications

Calculate each of the following.

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

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

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

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

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

Prove or disprove: ${\mathbb{Z}}_2[x]/⟨x^3+x+1⟩\cong {\mathbb{Z}}_2[x]/⟨x^3+x^2+1⟩\text{.}$

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

Prove that the ideal $⟨t+1⟩$ in $R_n$ is the code in ${\mathbb{Z}}_2^n$ consisting of all words of even parity.

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

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

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

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

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

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 $E\text{.}$

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.

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 $E=F(\alpha )\text{.}$

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

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

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

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 $E\cap F\text{?}$

Let $p$ be prime. Prove that $(p-1)!\equiv -1(\mathrm{mod}p)\text{.}$

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 $1\text{.}$

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.

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

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