Tutorial 4 : Finite Fields and Primitive Elements

1. Let $ K=\mathbb{F}{p^n}.$ Define $ f: K \rightarrow K$ by $ f(x)=x^2.$
Show that $ f$ is surjective if $ p=2.$
Show that the number of elements in $ f(K)=(p^n+1)/2.$
Let $ \alpha$ and $ \beta$ be nonzero elements of $ K$ Show that there exist $ x, y \in K$ such that                 $ \alpha x^2+\beta y^2 =-1,$

2. Show that the product of nonzero elements of a finite field is $ -1.$ Deduce Wilson's theorem: If $ p$ is a prime number then $ p\;\vert\;1+(p-1)!.$

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

4. Let $ K$ be a finite field with $ q$ elements. Define the zeta function

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$\displaystyle Z(t)=\frac{1}{1-t} \prod_{p} \frac{1}{1-t^{\deg p}}$

where $ p$ ranges over all monic irreducible polynomials over $ K.$ Prove that $ Z(t)$ is a rational function and determine this rational function.

5. Let $ \alpha=\sqrt[3]{2}, \zeta=(-1+\sqrt{-3})/2 $ and $ \beta=\alpha \zeta.$
Prove that for all $ c \in \mathbb{Q}, \gamma=\alpha+c\beta$ is a root of a sextic of the form $ x^6+ax^3+b.$
Prove that irr $ (\alpha+\beta,\mathbb{Q})$ is cubic.
     (c) Prove that irr $ (\alpha-\beta,\mathbb{Q})$ is sextic.

6. Construct infinitely many intermediate subfields of $ \mathbb{F}_p(u,v)/\mathbb{F}_p(u^p, v^p)$ where $ u,v$ are indeterminates.

7. Find a primitive element of $ \mathbb{F}_{2^{4}}$ over $ \mathbb{F}_2.$