Exercises XII

In the following exercises feel free to use the following facts.

Fact 1. Suppose that K is a field of characteristic zero and p a prime number. Let a be an element of K which is not a pth power in K. Then the polynomial X^{p} − a is irreducible over K.

Fact 2. Let L/K be a finite extension of degree d. Consider the map det : L → K which maps α ∈ L to the determinant of the K-linear map L → L, x ↦ αx. You check easily that det (sometimes called the norm map) satisfies the following two properties:

(a) det(αβ) = det(α) det(β), and

(b) det(α) = α^{d} if α ∈ K.

1. Let K, a ∈ K, and p be as in Fact 1. Let L = K(a^{1/p}). Compute det(a^{1/p}), where det is as in Fact 2. (Careful!)

2. Let K be a field of characteristic 0, and let p be an odd prime. Suppose that a is an element of K which is not a pth power in K. Show that X^{p^{2}} − a is irreducible over K. Hint: show that a^{1/p} is not a pth power in K(a^{1/p}).

3. Show by an example that the statement of exercise 2 is wrong if p = 2.

4. In this exercise K is a field with the following properties:

(a) The characteristic of K is 0,

(b) the algebraic closure K̄ of K is finite over K, and (c) K ≠ K̄.

Show that the finite group G = Gal(K̄/K) is a 2-group. Hint: Suppose G contains an element σ of odd prime order p in G. Show that the fixed field K̄^{<σ>} contains a primitive pth root of unity. Then use Kummer theory and Exercise 2.

Remark. This is the beginning of the proof that shows that fields as in Exercise 4 are special. Namely, it can only happen if [K̄ : K] = 2 and K is a “real closed field”.

5. Let p_{1} = 2, p_{2} = 3, p_{3} = 5, p_{4} = 7, etc be the prime numbers. Consider the group

∏_{n∈N} Z/p_{n}Z.

Show that any finite index subgroup is given by a sub-product

∏_{n∈S} Z/p_{n}Z.

for some subset S ⊂ N containing all but finitely many n ∈ N. (Feel free to use that the result holds for finite products of cyclic groups of pairwise distinct prime orders. But only if you actually understand what I mean by that...)

Warning. It isn’t true that any subgroup of the product is a “sub-product”.