Math 314  Homework Assignment 4  March 15, 2004

due date: March 23, 2004

1. Suppose that K ⊆ L is a finite Galois extension, and M_{1} and M_{2} any two intermediate fields (i.e., K ⊆ M_{i} ⊆ L for i = 1, 2). Let M be the smallest field in L containing both M_{1} and M_{2}, and let M′ = M_{1} ∩ M_{2}. (Another way to say this is that M′ is the largest subfield of L contained in both M_{1} and M_{2}).

If H_{1} and H_{2} are the subgroups of G = Aut(L/K) corresponding to M_{1} and M_{2} under the Galois correspondence, show that:

(a) The subgroup corresponding to M is H = H_{1} ∩ H_{2}, and

(b) the subgroup corresponding to M′ is the subgroup H′ generated by H_{1} and H_{2}.

2. Recall that a group G is a product G = H_{1} × H_{2} if and only if there are normal subgroups H_{1} ⊂ G and H_{2} ⊂ G such that H_{1} ∩ H_{2} = {e} and H_{1} · H_{2} (the subgroup generated by H_{1} and H_{2}) is equal to G.

Suppose that K ⊆ L is a finite Galois extension, and M_{1} and M_{2} are two intermediate fields such that:

(i) Both K ⊆ M_{1} and K ⊂ M_{2} are Galois extensions.

(ii) M_{1} ∩ M_{2} = K.

(iii) The smallest subfield of L containing both M_{1} and M_{2} is L itself.

If H_{1} and H_{2} are the subgroups of G = Aut(L/K) corresponding to M_{1} and M_{2} under the Galois correspondence, show that G = H_{1} × H_{2}.

Conversely, if the Galois group G is a product G = H_{1} × H_{2}, then show that there are two intermediate fields M_{1} and M_{2} having the properties above.

3. If K = ℚ and L = ℚ(α, ω, \sqrt{3}) with α = 2^{1/3} and ω = e^{2πi/3} then explain why K ⊂ L is a Galois extension and compute Aut(L/K).

You may use results (without having to reprove them) from previous homework problems in answering this question. Just be sure to mention explicitly which facts you’re using, and where you got them from.

4. Which of these two polynomials are irreducible over 𝔽_{5}?:

(a) x^{2} + x + 2

(b) x^{2} + x + 3 Which of these two polynomials are irreducible over 𝔽_{7}?:

(c) x^{2} + 5x − 2

(d) x^{2} + 5x − 3

5. For a quadratic polynomial f(x) = ax^{2} + bx + c, the number b^{2} − 4ac is called the discriminant of f(x).

(a) Compute the discriminants of the quadratic polynomials in question 4.

Note that two of the answers are numbers in 𝔽_{5} and two are numbers in 𝔽_{7}. A number y ∈ 𝔽_{p} is called a square if there is an x ∈ 𝔽_{p} with x^{2} = y.

(b) List the squares in 𝔽_{5} and 𝔽_{7}.

(c) Which of the discriminants are squares?

(d) What is the connection betwen the answer in (c) and the answers in question 4?

6. Suppose that L is a field with 244, 140, 625 elements.

(a) What is char(L)?

(b) If p = char(L), what is the index [L : 𝔽_{p}]?

(c) What is the Galois group Aut(L/𝔽_{p})?

(d) Draw the diagram of subfields of L and their inclusions.