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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 KummerDedekind 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.
 Fall '11
 Staff
 Number Theory

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