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8410Chapter2v2

# 8410Chapter2v2 - ABSOLUTE VALUES II TOPOLOGIES COMPLETIONS...

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Unformatted text preview: ABSOLUTE VALUES II: TOPOLOGIES, COMPLETIONS AND THE EXTENSION PROBLEM PETE L. CLARK Contents 2.1. Introduction and Reorientation 1 2.2. Reminders on metric spaces 5 2.3. Ultrametric spaces 7 2.4. Normed abelian groups 8 2.5. The topology on a normed field 10 2.6. Completion of a metric space 11 2.7. Completions of normed abelian groups and normed fields 12 2.8. Non-Archimedean Functional Analysis: page 1 15 2.9. Big Ostrowski Revisited 16 2.10. Theorems of Mazur, Gelfand and Tornheim 19 2.11. Proof of Theorem 3 Part I: Uniqueness 21 2.12. Proof of Theorem 5 22 2.13. Proof of Theorem 3 Part II: Existence Modulo Hensel-K¨ursch´ ak 25 2.14. Proof of Theorem 3 Part III: Krull Valuations 26 References 29 2.1. Introduction and Reorientation. In this chapter we will study more explicitly the topology on a field induced by a norm. Especially interesting from this perspective are the (nontrivially) normed fields which are locally compact with respect to the norm topology. But we have been studying normed fields for a little while now. Where are we going? What problems are we trying to solve? Problem 1: Local/Global Compatibility. Arguably the most interesting re- sults in Chapter 1 were the complete classification of all norms on a global field K , i.e., a finite extension of either Q (a number field) or F q ( t ) for some prime power q (a function field). We interrupt for two remarks: Remark 1: Often when dealing with function fields, we will say “Let K/ F q ( t ) be Thanks to John Doyle and David Krumm for pointing out typos. Section 2.9 on Big Ostrowski was written following lecture notes of David Krumm. 1 2 PETE L. CLARK a finite separable field extension”. It is not true that every finite field extension of F q ( t ) is separable: e.g. F q ( t 1 q ) / F q ( t ) is an inseparable field extension. However, the following is true: if ι : F q ( t ) , → K is a finite degree field homomorphism – don’t forget that this wordier description is the true state of affairs which is being elided when we speak of “a field extension K/F ” – then there is always another finite degree field homomorphism ι ′ : F q ( t ) , → K which makes K/ι ′ ( F q ( t )) into a separable field extension: e.g. [Eis, Cor. 16.18]. Remark 2: In the above passage we could of course have replaced F q ( t ) by F p ( t ). But the idea here is that for an arbitrary prime power q , the rational function field F q ( t ) is still highly analogous to Q rather than to a more general number field. For instance, if K is any number field, then at least one prime ramifies in the extension of Dedekind domains Z K / Z . However, the extension F q [ t ] / F p [ t ] is everywhere un- ramified. Moreover, F q [ t ] is always a PID....
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8410Chapter2v2 - ABSOLUTE VALUES II TOPOLOGIES COMPLETIONS...

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