8410Chapter4

8410Chapter4 - STRUCTURE THEORY OF LOCAL FIELDS PETE L....

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Unformatted text preview: STRUCTURE THEORY OF LOCAL FIELDS PETE L. CLARK Contents 4. Structure Theory of CDVFs 1 4.1. Serre’s Kummer-Dedekind Criterion 3 4.2. Unramified extensions 3 4.3. Totally ramified extensions and Eisenstein’s Criterion 4 4.4. Tamely ramified extensions 5 4.5. Wildly ramified extensions 6 4.6. More on the PGM Filtration 6 4. Structure Theory of CDVFs We now specialize to the following situation: let ( K, | | ) be a complete, non- Archimedean field whose valuation ring R is a DVR and whose residue field k is perfect. Under these hypotheses we can give a much more penetrating analysis of the structure of the absolute Galois group g K = Gal( K sep /K ) and also of the multiplicative group K × . Recall that a finite extension L/K is unramified if e ( L/K ) = 1; equivalently, f ( L/K ) = [ L : K ]. (Note that we are using our assumption of the perfection of k here, for otherwise we would need to add the condition that the residual extension l/k is unramified.) An algebraic extension L/K is unramified if all of its finite subextensions are unramified. A finite extension L/K is totally ramified if e ( L/K ) = [ L : K ]; equivalently, l = k . An algebraic extension L/K is totally ramified if each finite subextension if totally ramified; equivalently, l = k . Let p be the characteristic exponent of the residue field k . (In other words, when k has positive characteristic, we take p to be the characteristic; when k has char- acteristic 0, we take p = 1.) Here is a new definition: a finite extension L/K is tamely ramified if e ( L/K ) is prime to p . An algebraic extension is tamely ramified if every finite subextension is tamely ramified. Note that in particular every unramified extension is tamely ram- ified, so perhaps more accurate terminology would be “at worst tamely ramified”, but the terminology we have given is standard. Note also that if char( k ) = 0 then every algebraic extension of K is tamely ramified. 1 2 PETE L. CLARK An extension L/K is totally tamely ramified , or TTR , if it is both totally ramified and tamely ramified. Both unramified and tamely ramified extensions are distinguished classes of field extensions in the sense of Lang, as we now explain. A class of field extensions C = { L/K } is said to be distinguished if it satisfies the following two conditions: (DE1) (Tower condition): if K/F and L/K are both in C , then L/F is in C . (DE2) (Base change condition): suppose E,F,K are subfields of a common field, and F ⊂ K , F ⊂ E and K/F ∈ C . Then EK/E ∈ C . Exercise 4.1: Show that from (DE1) and (DE2) we have the following formal con- sequence: (DE3) Suppose K,L 1 ,L 2 are subfields of a common field, with K contained in both L 1 and L 2 and that L 1 /K, L 2 /K ∈ C . Then L 1 L 2 /K ∈ C ....
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This note was uploaded on 10/26/2011 for the course MATH 8410 taught by Professor Staff during the Fall '11 term at UGA.

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8410Chapter4 - STRUCTURE THEORY OF LOCAL FIELDS PETE L....

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