Carolina M
Posted on Tuesday, 06 May, 2003 - 05:38 am:
Please help, am having problems knowing how to start these questions

Show that if F is a finite field, and 'n' is in the positive integers, then there exists an irreducible polynomial f in F[t] of degree n.

Suppose that K/F is algebraic. Suppose that char F =/= 2 (does not equal 2), & for all 'a' in K the minimal polynomial m\_F(A) has degree at most 2. Show that if 'a' in K is fixed by every F-automorphism sigma: K-> K then 'a' is in F (K/F need not be finite) What if char F =2 ?
Demetres Christofides
Posted on Tuesday, 06 May, 2003 - 09:19 am:
For the first part there must be a more elegant way, but the only one I can think of is the following:

1) Show that the product of all the irreducible polynomials of degree dividing n is Xqn - X.

2) Let an be the number of irreducible polynomials of degree n. Show that

å
d|n 
 d ad = qn.

3) If there are no irreducible polynomials of degree n, derive a contradiction.

For the second part, suppose a is in K but not F. Take its minimal polynomial f. It cannot be linear, so it is of degree 2. So f(x) = (x-a)(x-b) for some b with ab, a+b in F. Can you now find an F-automorphism of K which does not fix a? Whatcan go wrong ifchar(F) = 2? Can you find a counterexample? Demetres