## Tutorial 4 : Finite Fields and Primitive Elements

1. Let Define by
(a)
Show that is surjective if
(b)
Show that the number of elements in
(c)
Let and be nonzero elements of Show that there exist such that

2. Show that the product of nonzero elements of a finite field is Deduce Wilson's theorem: If is a prime number then

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

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

where ranges over all monic irreducible polynomials over Prove that is a rational function and determine this rational function.

5. Let and
(a)
Prove that for all is a root of a sextic of the form
(b)
Prove that irr is cubic.
(c) Prove that irr is sextic.

6. Construct infinitely many intermediate subfields of where are indeterminates.

7. Find a primitive element of over