MODERN ALGEBRA (MATH 210) PROBLEM SET 7

1. Prove that ℚ(π) ≅ ℚ(x). You may use the fact that π is transcendental.

2. Suppose q is a prime power. Show that there are (q^{3} − q)/3 irreducible monic degree 3 polynomials in 𝔽_{q}[x]. (Hint: Consider the elements of 𝔽_{q^{3}}, and their minimal polynomials over q.) How many irreducible monic degree 12 polynomials are there in 𝔽_{q}[x]?

3. Let E be the field k(x). Consider the six automorphisms of E given by mapping f(x) to f(x), f(1 − x), f(1/x), f(1 − 1/x), f(1/(1 − x)), f(x/(x − 1)) respectively. Show that these automorphisms form a group. Let F be the fixed point field. Show that I = (x^{2} − x + 1)^{3}/x^{2}(x − 1)^{2} ∈ F. Show that F = k(I) and [E : F] = 6. (Hint: Find a sixth degree equation with coefficients in k(I) satisfied by x. Use a useful theorem from class.) Hence k(x)/k(I) is a Galois extension. What is its Galois group?

4. Suppose p_{1}, …, p_{n} are distinct prime numbers, and let F = ℚ(\sqrt{p_{1}}, …, \sqrt{p_{n}}). Find the degree of the extension F/ℚ. Show that F/ℚ is Galois, and find its Galois group.

5. Suppose e_{1}, …, e_{n} are the elementary symmetric functions in the n variables x_{1}, …, x_{n} over some field k, i.e. e_{1} = x_{1} + … x_{n}, e_{2} = x_{1}x_{2} + x_{1}x_{3} + ⋯ + x_{n−1}x_{n}, …, e_{n} = x_{1}x_{2} … x_{n}. Let p_{i} = x_{1}^{i} + ⋯ + x_{n}^{i} (i a positive integer). By the theorem of symmetric functions, p_{i} is a polynomial in the e_{j}. Show this explicitly as follows. We have seen that the polynomial f(T) ∈ (k(x_{1}, …, x_{n}))[T]

f(T) = T^{n} − e_{1}T^{n−1} + e_{2}T^{n−2} − ⋯ + (−1)^{n−1}e_{n−1}T + (−1)^{n}e_{n}

has roots x_{1}, …, x_{n}. Rewrite x_{1}^{a}f(x_{1}) + ⋯ + x_{n}^{a}f(x_{n}) = 0 in terms of p_{a}, …, p_{a+n} (a a positive integer), and show that if p_{i} is a polynomial in the e_{j} for i < a + n, then p_{a+n} is too. Deal also with the cases where 1 ≤ i ≤ n.

6. Suppose E/F is a Galois extension of degree n with Galois group {σ_{1}, …, σ_{n}}, and a_{1}, …, a_{n} is a basis for E over F. Show that

det ( \begin{matrix}σ_{1}(a_{1}) & σ_{1}(a_{2}) & ⋯ & σ_{1}(a_{n}) \\ σ_{2}(a_{1}) & σ_{2}(a_{2}) & ⋯ & σ_{2}(a_{n}) \\ ⋮ & ⋮ & ... & ⋮ \\ σ_{n}(a_{1}) & σ_{n}(a_{2}) & ⋯ & σ_{n}(a_{n}) \end{matrix} )^{2}

is a non-zero element of F. (Hint: Show that it is non-zero separately. Recall the proof that if {σ_{1}, …, σ_{n}} is a group of automorphisms of a field E′, and F′ is the fixed field, then [E′ : F′] = n. The proof used the fact that there could be no nontrivial solution to a certain linear equation.)

The set is due Tuesday, December 3 at 3:30 pm in Pierre Albin’s mailbox.

Date: Tuesday, November 26, 2002.