Math 254A, UC Berkeley, Fall 2001 (Kedlaya) Problem Set 9: Ramification in Galois Extensions Due in class Monday, November 5

Please do six of the following problems, including the first problem.

Leftover Problem about Ramification.

1. Let K be a finite extension of ℚ_{p} and L/K be a tamely ramified (but not necessarily totally ramified) extension. Compute the discriminant Disc(L/K) in terms of e(L/K) and f(L/K). (The point: the discriminant only gets really nasty in the wildly ramified case.)

Ramification in Galois Extensions.

1. Neukirch exercise I.9.1: if L/K is a Galois extension of number fields with noncyclic Galois group, then there are at most finitely many totally nonsplit prime ideals of K (i.e., primes of 𝔬_{K} with only one prime of 𝔬_{L} over them). Also, use this argument to explicitly construct an irreducible polynomial over ℤ which is not irreducible modulo any prime.

2. Neukirch exercise I.9.2 (corrected): if L/K is a Galois extension of number fields and 𝔟 is a prime ideal which is unramified over L (i.e., for 𝔭 = 𝔟 ∩ K, e(𝔟/𝔭) = 1), there exists a unique automorphism ϕ_{𝔟} of L over K such that ϕ_{𝔟}(a) ≡ a^{q} (mod 𝔟), where q = Norm(𝔭). It is called the Frobenius automorphism of 𝔟; observe also that ϕ_{𝔟} generates the decomposition group of 𝔟 in this case.

3. Neukirch I.9.3: let L/K be an extension of prime degree p with solvable normal closure. If the unramified prime ideal 𝔭 in L has two prime factors 𝔟 and 𝔟′ of inertia degree 1, then it is totally split. (Hint from Neukirch: use a theorem of Galois, that if G is a transitive solvable permutation group of prime degree p, then there is no nontrivial permutation of G that fixes two distinct letters.)

4. Neukirch I.9.4: Let L/K be a finite, but not necessarily Galois, extension of number fields and N/K the normal closure of L/K. Show that a prime ideal 𝔭 of K is totally split in L if and only if it is totally split in N. (Hint from Neukirch: use the double coset decomposition H\G/G_{𝔟}, where G = G(N/K), H = G(N/L), and G_{𝔟} is the decomposition groups of a prime ideal 𝔟 over 𝔭.)

5. Prove that the previous exercise remains true if “totally split” is replaced by “unramified” or by “tamely ramified”. Optional: describe more generally how to compute the ramification and inertia degrees of the primes of L above 𝔭 in terms of the double coset decomposition.

6. Let P (x) be a polynomial of degree n over 𝔬_{K} for some number field K such that modulo some prime 𝔭, P (x) is irreducible, while modulo some other prime 𝔮, P (x) factors as an irreducible quadratic times pairwise coprime linear factors. Prove that the Galois group of the normal closure of K[x]/(P (x)) is the full symmetric group S_{n}. (Hint: use Frobenius automorphisms.)

7. Formulate a sensible definition of “the decomposition/inertia group of an archimedean embedding of a number field”. As a test, your definition should allow the proof of the preceding problem to pass through unchanged to prove the following: if P (x) is a polynomial over ℤ which is irreducible modulo some prime p and which has exactly two nonreal roots, then the Galois group of the normal closure of ℚ[x]/(P (x)) is the full symmetric group S_{n}.