Math 254A, UC Berkeley, Fall 2001 (Kedlaya) Lecture 24: Ramification in Galois Extensions October 29, 2001

Reference. Neukirch, Section 1.9; optionally, Section 2.9 as well. Jargon watch. Given an extension L/K of fields, the normal closure of L/K is the compositum of the image of L′ under all elements of Gal(\overline{K}/K). If this equals L, we say L/K is normal. We say L/K is Galois if it is finite, normal and separable, or equivalently, if # Gal(L/K) = [L : K]. Note that finite and separable are automatic for L/K an extension either of number fields or of finite extensions of ℚ_{p}. Outline of lecture.

1. Let L/K be a Galois extension of number fields with Galois group G. Show that G acts transitively on the primes 𝔟 of 𝔬_{L} lying above a prime 𝔭 of 𝔬_{K}. Note in particular that they all have the same ramification index and inertia degree over 𝔭.

2. Let 𝔟 be a prime of 𝔬_{L} lying over the prime 𝔭 of 𝔬_{K}. Define the decomposition group G_{𝔟} of 𝔟 over K as the set of σ ∈ G mapping 𝔟 to itself (but not necessarily by the identity map!). Define the fixed field of G_{𝔟} as the decomposition field of 𝔟 over K.

3. Interpret some of the splitting behavior of 𝔭 in terms of the decomposition group. E.g., G_{𝔟} = 1 if and only if 𝔭 is totally split.

4. Show that G_{𝔟} surjects onto the Galois group Gal((𝔬_{L}/𝔟)/(𝔬_{K}/𝔭). The kernel of this map is the inertia group I_{𝔟} of 𝔟 over K, whose fixed field is the inertia field of 𝔟 over K. Again, interpret splitting behavior in terms of this: the order of I_{𝔟} is the ramification index e(𝔟/𝔭), and the order of G_{𝔟}/I_{𝔟} is the inertia degree.

5. Note that G_{𝔟} can also be interpreted as the Galois group of the extension L_{𝔟}/K_{𝔭} of local fields, and that its subgroup I_{𝔟} is the Galois group of L_{𝔟} over its maximal unramified subextension (over K_{𝔭}). (Of course, one can also define the inertia subgroup for any Galois extension of local fields.)

6. Point out that finite Galois extensions of local fields are always solvable, so that decomposition groups are always solvable. More precisely, the decomposition group is the semidirect product of the inertia group by the Galois group of the residue field extension, the inertia group is the semidirect product of the “wild inertia group” by a cyclic group, and the wild inertia group is a p-group, where p is the characteristic of the residue field.

7. If time permits, mention how this all can be interpreted for a non-Galois extension. Namely, if L/K is not Galois, let N/K be its normal closure, put G = Gal(N/K), and put H = Gal(N/L). Now for 𝔟 a prime of N above 𝔭, the double cosets H\G/G_{𝔟} correspond to the primes of L above 𝔟, and one can relate the splitting of 𝔭 to this decomposition: for examples, see the homework.