## Tutorial 3 : Separable Extensions and Finite Fields

1. Let where is a positive integer and is a field of characteristic Show that is irreducible if and only if

2. Show that is the largest perfect subfield of

3. Identify the finite fields and

4. Find a necessary and sufficient condition on and so that is a subfield of

5. Draw the poset of subfields of

6. Show that the order of the Frobenius automorphism is

7. Let be irreducible over and let it have a root Show that the roots of are precisely

8. Let and be subfields of having and elements respectively. How many elements does the field have ?