Finite Fields

Find all the zeroes of $f(x)=x^{4}+x+1$ over the field $F$ described in 1.6, using $\alpha$ as one possible zero of $f(x)$ in $F$.

Prove 1.9.

Consider the finite field $F$ in 1.7. What are its subfields?

Is

Justify your answer using a concise argument.

Let $\alpha$ be a zero of $x^{3}+x^{2}+1$ in some extension field of $\mathbb{Z}_{2}$.

1. Find $(\alpha^{2}+\alpha+1)^{-1}$ in $\mathbb{Z}_{2}(\alpha)$.

2. Hence solve the equation

   for $x$ over $\mathbb{Z}_{2}(\alpha)$.

(Hint: refer to the conversion formulae given in 1.7.)

Explain why every element of a finite field of characteristic $p$ can be written in the form $a^{p}$, where $a$ is an element of the same finite field.(Hint: Consider the proof of 1.51.)

Consider the irreducible polynomial $f(x)=x^{2}+2x+2$ over $\mathbb{Z}_{3}$. Let $\alpha$ be a zero of $f(x)$ in some extension field of $\mathbb{Z}_{3}$. Find the other zero(es) of $f(x)$ in $\mathbb{Z}_{3}(\alpha)$.

Let $p(x)=x^{3}+2x+2\in\mathbb{Z}_{3}[x]$.

1. Show that $p(x)$ is irreducible over $\mathbb{Z}_{3}$.

2. Let $\alpha$ be a zero of $p(x)$ in the field $F=\mathbb{Z}_{3}[x]/\left\langle p(x)\right\rangle$. Show that $\alpha$ is not a generator of $F^{\ast}$.

3. Find a generator of $F^{\ast}$, and prove that your answer is indeed a generator of $F^{\ast}$. Express your answer in terms of $\alpha$.

Prove that any irreducible factor of $x^{32}-x$ over $\mathbb{Z}_{2}$ has a degree of 1 or 5.

Prove that every finite extension of a finite field is simple.

Prove that no finite field is algebraically closed.

Let $F$ be a field. We say that $a\in F$ is a square if and only if there exists a $b\in F$ such that $a=b^{2}$. If $a\in F$ is not a square it is called a non-square.

Now let $p$ be an odd prime. Show that if any non-square in $\mathrm{GF}(p)$ is still a non-square in $\mathrm{GF}(p^{n})$, then $n$ is an odd positive integer.(Hint: consider the polynomial $f(x)=x^{2}-a$, where $a$ is a non-square.)