8410Chapter5

8410Chapter5 - LOCALLY COMPACT FIELDS PETE L CLARK Contents...

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Unformatted text preview: LOCALLY COMPACT FIELDS PETE L. CLARK Contents 5. Locally compact fields 1 5.1. The classification of nondiscrete locally compact topological fields 1 5.2. Roots of unity in locally compact fields 4 5.3. The higher unit groups 5 5.4. The number of n th power classes in a locally compact field 6 5.5. The number of degree m extensions of a locally compact field 8 5.6. Pontrjagin duality 9 5.7. Additive autoduality of locally compact fields 9 5. Locally compact fields 5.1. The classification of nondiscrete locally compact topological fields. Some of the most important theorems in mathematics give complete classifications of certain fundamental structures. Examples: the classification of (topological!) surfaces, the classificaiton of simple Lie algebras, the classification of finite simple groups. In this section we discuss a classification theorem which belongs somewhere in the above pantheon. Theorem 1. Let L be a locally compact, nondiscrete topological field. a) Then L is a finite extension of one of the following fields: (i) K = R . (ii) K = Q p . (iii) K = F p (( t )) . b) In case (i) L = R or L = C . c) In case (ii) the ramification index e ( L/ Q p ) and residual degree f ( L/ Q p ) are uniquely determined by the abstract field L , and for any given e,f ∈ Z + , the number of finite extensions L/ Q p of ramification index e and residual degree f is finite and nonempty. d) In case (iii) the residual degree f is determined by the abstract field L , but the ramification index is not. Moreover, every totally ramified extension of F q (( t )) is isomorphic to F q (( t )) . To prove this result in full requires some nontrivial tools from the theory of topolog- ical groups: namely, the existence of Haar measure. We will give a rather superficial discussion of this later on, not because it is necessary for what we wish to do later c Pete L. Clark, 2010. 1 2 PETE L. CLARK in our course, but because it is interesting and natural and deserves to be part of our general mathematical culture. So we begin by simplifying things. Let us restrict our attention to discretely valued, non-Archimedean fields and classify all locally compact fields among them. First a simple lemma of functional analysis type. Lemma 2. Let ( K, | | ) be a locally compact normed field, and let ( V, || || ) be a normed K-space. Then V is locally compact iff dim K V is finite. Proof. ... Theorem 3. Let ( K,v ) be a discretely valued 1 non-Archimedean field, with valua- tion ring R and residue field k . TFAE: (i) K is locally compact. (ii) K is ball compact. (iii) R is compact. (iv) K is complete, and the residue field k is finite. (v) K may be expressed as a finite extension of Q p or of F p (( t )) , for a suitable prime number p ....
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8410Chapter5 - LOCALLY COMPACT FIELDS PETE L CLARK Contents...

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