HW due May 4

1. Let F be Galois over K, and f in F be algebraic over K. Let {f_i} be the orbit of f under the Galois group.
a. Show that the set {f_i} is finite, and
b. the polynomial \Prod (x-f_i) is in K[x], irreducible, and has f as a root.

2. Let S_n act on the field Q(a₁,...,a_n) of rational functions in n variables. Let e_i^S be the ith elementary symmetric polynomial in the {a_j, j in S}. As that Wikipedia page (v. 4/28/07) indicates, the invariant subfield is generated by the {e_i^1,2,...,n =: e_i}.

a. What is the minimal polynomial of a₁, in Q(e₁,...,e_n)?
b. Let n=4. Describe all subgroups of S₄ up to conjugacy.
For each conjugacy class of subgroup of S₄, write down generators for the fixed field.
For each such field F, write down the minimal polynomial in F[x] for each a_i.

3. Let F be a field of characteristic p.
a. If F is finite, show that the Frobenius is onto.
b. If F is algebraically closed, show that the Frobenius is onto.
c. Give an example where it's not onto.