8410Chapter3 - ABSOLUTE VALUES III: THE FUNDAMENTAL...

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Unformatted text preview: ABSOLUTE VALUES III: THE FUNDAMENTAL IN/EQUALITY, HENSEL AND KRASNER PETE L. CLARK Contents 3. Residual degree and ramification index 1 3.1. Hensels Lemma 5 3.2. Squares in local fields 6 3.3. Quadratic forms over local fields 6 3.4. Roots of unity in local fields 9 3.5. Krasners Lemma and Applications 12 3.6. Multi-complete and multi-Henselian fields 13 References 17 3. Residual degree and ramification index Let ( K,v ) be a valued field, with valuation ring R and maximal ideal m . As with any maximal ideal, the quotient ring R/ m is a field, called the residue field of K and denoted k . (Note that we have switched from k to K for our normed/valued field so as to allow the introduction of the residue field.) We have a canonical sur- jective map R k called the reduction map . Example: Suppose K is any field and v is the trivial (i.e., identically zero) val- uation. Then R = K , m = 0, so k = K and the reduction map is an isomorphism. Conversely, if the reduction map is injective, the valuation is trivial. Example: Let ord p be the p-adic norm on Q . Then the valuation ring is the local ring R of all rational numbers of the form a b with b not divisible by p . This is of course the localization of Z at the maximal ideal ( p ). It follows that R/ m = Z / ( p ) = F p . Example: Let k be a field and R = k [[ t ]] and K = k (( t )). Then the maximal ideal consists of all formal power series with 0 constant term, and it is easily seen that R/ m = k in such a way that the composite map k , R R/ m = k is the identity. Thus in this case the residue field is also realized as a subfield of K . Example: let k be a field, R = k [ t ], K = k ( t ). Let ord t be the valuation cor- responding to the prime element ( t ). Then again the residue field is isomorphic to R/ ( t ) = k . More generally: Date : June 6, 2010. 1 2 PETE L. CLARK Proposition 1. Let R be a Dedekind domain with fraction field K . Let p be a nonzero prime ideal of R , and let v = ord p be the p-adic valuation. Then the residue field is naturally isomorphic to R/ p . Exercise 3.1: Prove Proposition 1. Now let ( L,w ) / ( K,v ) be an extension of valued fields. Recall that this means that we have a field homomorphism : K , L such that w = v . In such a situation, induces an embedding of valuation rings R , S and of maximal ideals m R , m S . We may therefore pass to the quotient and get a homomorphism : k = R/ m R , S/ m S = l, called the residual extension . The degree [ l : k ] is called the residual degree and is also denoted f ( L/K ). Exercise 3.2: Suppose that L/K is algebraic. Show that l/k is algebraic. Again let ( K,v ) , ( L,w ) be a homomorphism of valued fields. Then we have v ( K ) w ( L ). We define the ramification index e ( L/K ) to be [ w ( L ) : v ( K )]. In terms of the associated norms, we have e ( L/K ) = | L | | K | ....
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8410Chapter3 - ABSOLUTE VALUES III: THE FUNDAMENTAL...

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