1. Let F be a field with p^n elements. Show that F has a subfield K with p^m elements if

and only if m | n.

2. Let K be a finite field. Show that the product of all the nonzero elements of K is −1.

3. Write f(x) = x^16 − x in Z2[x] (that is, f(x) = x^2^4− x) as the product of all monic

irreducible polynomials over Z2 of degree dividing 4.

4. (a) Determine the number of monic irreducible polynomials of degree 6 in Z2[x].

(b) Determine the number of monic irreducible polynomials of degree 11 in Z3[x].

Ideas:

1.I think this may have something to do with degrees

2.I started by considering |k|=p^n. Then I tried looking at x^p^n-x=x(x^(p^n-1)-1).

3.

4. Consider degrees?