Math 254A, UC Berkeley, Fall 2001 (Kedlaya) Final Examination Due in my mailbox Monday, December 10 by 5 PM

Instructions. Please submit solutions to at least eight of the following problems; your best eight problems will be counted. You may:

• Consult Neukirch, your own lecture notes, any handouts distributed in class, your graded problem sets, or the Magma documentation.

• Invoke the statement of any result from Neukirch, from lecture, or from the problem sets (submitted or not) without proof, as long as you specify where it came from.

• Consult me or Frank with questions.

• Use Magma (or a computer algebra system of your choice) where specified.

You may not:

• Discuss the exam questions with anyone other than me or Frank, until after you have submitted the exam.

• Use Magma or other computational device where not directed to do so.

• Consult any reference, print or electronic, other than those mentioned above.

Problems.

Throughout the problems, let ζ_{n} denote a primitive n-th root of unity.

1. (a) Prove that the field ℚ(ζ_{13}) contains a unique subfield K of degree 4 over ℚ, generated by x = ζ_{13} + ζ_{13}^{3} + ζ_{13}^{9}.

(b) Determine the minimal polynomial and the different of x. You may use Magma for this and the next part.

(c) Exhibit an element of 𝔬_{K} not contained in ℤ[x]. (Hint: compute the discriminant of the minimal polynomial of x first to find a square factor that you can remove.)

2. Let α be an algebraic integer all of whose conjugates in ℂ are real and lie in the interval [−2, 2]. Prove that α = 2 cos(2πr) for some r ∈ ℚ. (Hint: reduce to Kronecker’s theorem from Problem Set 2.)

3. (a) Show that the class number of ℤ[\frac{1}{2}(1 + \sqrt{−11})] is 1. (Hint: use the bounds from Problem Set 5 to reduce this to a finite computation.)

(b) Find all integers x, y ∈ ℤ such that y^{2} = x^{3} − 11.

4. Find all solutions of the equation

x^{3} + 5y^{3} + 25z^{3} − 15xyz = ±1

with x, y, z ∈ ℤ. You may use Magma to search for some small solutions. (Hint: relate this to the unit group of ℚ(\sqrt[3]{5}), whose ring of integers you may assume is ℤ[\sqrt[3]{5}] without proof.)

5. Let K be a finite unramified extension of ℚ_{p} with valuation ring 𝔬_{K}, and let 𝔪_{K} be the maximal ideal of 𝔬_{K}. Let σ be the unique automorphism of K such that x^{σ} ≡ x^{p} (mod 𝔪_{K}). Prove that for any x ∈ 𝔬_{K} such that x ≡ 1 (mod 𝔪_{K}), there exists y ∈ 𝔬_{K}^{∗} such that y^{σ}/y^{p} = x. (Hint: imitate one proof of Hensel’s lemma.)

6. Determine the number and degrees of the irreducible factors of the polynomial x^{4} + 9x^{2} − 2 over ℚ_{2}. (Hint: make sure to apply Hensel’s lemma only when valid.)

7. Let L be a number field whose absolute discriminant is squarefree and let K be a subfield of L. Prove that K = ℚ. (Hint: use properties of the relative discriminant to show that K/ℚ is unramified.)

8. Let P (x) = x^{6} − x^{5} − x^{4} + x + 1. You may assume P (x) is irreducible over ℤ.

(a) Compute the discriminant D of P (x) using Magma, and check that it is 3^{3} times a squarefree number.

(b) Let K be the normal closure of ℚ[x]/(P (x)). Prove that K is unramified over all primes of ℚ(\sqrt{D}) not above (3).

(c) Prove that K is also unramified over the prime of ℚ(\sqrt{D}) above (3). (Hint: determine the decomposition and inertia groups for a prime of K above (3).)

9. Let n be a positive integer which is not a prime power. Prove that 1 − ζ_{n} is a unit in ℤ[ζ_{n}]. (Hint: compute ∏_{i=1}^{n−1}(1 − ζ_{n}^{i}) and show that the terms where i is not coprime to n account for all of the prime factors. For starters, try the case n = pq.)

10. (a) Prove that the field ℚ(ζ_{31}) contains a unique subfield K of degree 5 over ℚ. (b) Prove that 2 is unramified and splits completely in K.

(c) Use (b) to prove that there does not exist α ∈ 𝔬_{K} such that 𝔬_{K} = ℤ[α]. (Hint: derive a contradiction from the splitting criterion that holds when 𝔬_{K} = ℤ[α].)

11. Neukirch exercise III.2.1:

(a) Let L/K be finite extensions of ℚ_{p}. Prove that Disc(L/K) is generated by Disc(1, α, …, α^{n−1}), where α runs over all elements of 𝔬_{L} such that K(α) = L.

(b) Prove that this statement does not hold for number fields, by showing that it fails for K = ℚ and L = ℚ[x]/(P (x)), where P (x) = x^{3} − x^{2} − 2x − 8. You may use without proof the fact that the discriminant of P is −2012. (Hint: show that 1, x, (x^{2} + x)/2 is an integral basis of L.)