Math 254A, UC Berkeley, Fall 2001 (Kedlaya) Problem Set 8: Ramification and Splitting Due in class Monday, October 29

Please submit six of the following problems.

Extension of Valuations.

1. Neukirch exercise II.8.3: How many extensions to ℚ(\sqrt[n]{2}) does the archimedean absolute value | · | of ℚ admit?

Ramification and Splitting.

1. Neukirch I.8.3: If a prime ideal 𝔭 of a number field K is totally split in the extensions L/K and L′/K, then it is also totally split in the compositum LL′/K.

2. Write a Magma program to determine the splitting behavior (i.e., the number of prime factors and the corresponding pairs (e, f)) of primes of ℚ in the ideal ℚ(\sqrt[3]{2}). (Hint: relate the splitting to the factorization of the polynomial x^{3} − 2 modulo p.) Optional: compute some examples and look for patterns in the data; it may help to separate primes based on p (mod 3).

Interlude on Quadratic Reciprocity.

1. For n = ∏_{i} p_{i}^{e_{i}} odd and coprime to a, define the Legendre symbol (\frac{a}{n}) as ∏_{i} (\frac{a}{p_{i}} )^{e_{i}}. Formulate a version of quadratic reciprocity (Theorem 8.6) for this generalized Legendre symbol, and use this to describe a version of the Euclidean algorithm to compute Legendre symbols without factoring into primes.

2. Neukirch II.8.8: Let a_{n} = \frac{ϵ^{n}−ϵ^{′n}}{\sqrt{5}} , where ϵ = \frac{1+\sqrt{5}}{2}, ϵ′ = \frac{1−\sqrt{5}}{2}, so that a_{n} is the n-th Fibonacci number. If p is a prime number other than 2 or 5, prove that a_{p} ≡ (\frac{p}{5}) (mod p).

The Relative Discriminant.

1. Neukirch defines the relative discriminant of an extension L/K of number fields as the ideal generated by Disc(α_{1}, …, α_{n}) for all choices α_{1}, …, α_{n} of a basis of L/K consisting of elements of 𝔬_{L}. Show that this coincides with the definition I gave in class. (Hint: you might want to peek ahead to Section III.2.)

2. If L/K is an extension of number fields, prove that the ideal (Disc(L)/ Disc(K)) (the quotient of the absolute discriminants) is equal to the absolute norm of the relative discriminant Disc(L/K). (Hint: use the definition of the relative discriminant from class.)

3. Prove that if P (x) is an irreducible monic polynomial over ℚ whose discriminant D is squarefree, then ℚ[x]/(P (x)) is unramified over ℚ(\sqrt{D}).

4. Compute the relative discriminant Disc(ℚ(ζ_{p})/ℚ), using the fact that the valuation ring of ℚ_{p}(ζ_{p}) equals ℤ_{p}[ζ_{p}], proved in class (and in Section II.7 of Neukirch). Use this result to prove that the ring of integers of ℚ(ζ_{p}) equals ℤ[ζ_{p}]. Optional: peek ahead to Section I.10 and compare the proof given there.