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

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

(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.

(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