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Unformatted text preview: Math 676. Some examples of extension fields The purpose of this handout is to probe the hypotheses of some of our results in class concerning the structure of finite extensions of local fields, or more generally of fraction fields of complete discrete valuation rings. 1. Number of extensions of a local field In class we saw that if K is a local field and n is a positive integer not divisible by char( K ) then the set of K-isomorphism classes of degree- n extensions of K is a finite set. Recall that the condition char( K )- n is crucial in the proof, as otherwise the compact space of Eisenstein polynomials over K with degree n/f (for a fixed f | n ) contains some inseparable polynomials (if char( K ) | ( n/f )) and so the locus of separable Eisenstein polynomials of degree n/f would appear to be probably non-compact (as it is the complement of a hypersurface in a compact space, the hypersurface being the zero-locus of the universal discriminant for monic polynomials of degree n/f ). Let us show that this apparent non-compactness for char( K ) | n is genuine and in fact leads to coun- terexamples to the desired finiteness statement in such cases. More specifically, we fix a local field K with characteristic p > 0 and we seek to show that there are infinitely many totally ramified separable p-extensions of K (up to K-isomorphism). Iterating such a construction would give infinitely many totally ramified sep- arable p r-extensions (up to K-isomorphism) for any r > 0, and then forming (linearly disjoint) composites with a totally tame extension K ( 1 /e ) for any e > 0 not divisible by p (and for a fixed choice of uniformizer K ) would give rise to infinitely many totally ramified separable extensions of K (up to K-isomorphism) with any desired degree n divisible by p . To carry out the construction of the infinitely many non-isomorphic totally ramified separable p-extensions of K , note that we do not really need to keep track of ramification: a separable p-extension K /K either has e ( K | K ) = p or e ( K | K ) = 1. In the first case K /K is totally ramified, and in the second case it is an unramified p-extension. There is only one unramified extension of each degree (due to the Galois theory of finite fields), and removing one element from an infinite list still yields an infinite set. Thus, we focus on the following problem: construct infinitely many separable p-extensions K /K that are pairwise non-isomorphic over K . (Note that it is crucial to emphasize the K-isomorphism aspect, since if we ignore the K-structure and consider mere abstract topological field isomorphism then everything will have to collapse: any local field is abstractly topologically isomorphic to a formal Laurent series field over its residue field and so is determined up to abstract (topological) field isomorphism by its residue field alone.) We will make examples that are even Galois (cyclic) of degree p , via ArtinSchreier theory. Recall the main result of ArtinSchreier theory: for any field...
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