By J. W. S. Cassels, A. Frohlich
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Additional info for Algebraic Number Theory: Proceedings of an Instructional Conference Organized by the London Mathematical Society (A Nato Advanced Study Institute W)
Appendix C. Hensel’s Lemma . . . . . ;: 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 44 45 46 47 48 49 52 53 56 58 60 62 63 67 68 70 71 73 76 80 . . 83 1. Valuations We shall be concerned only with rank 1 valuations, so for brevity, valuation will mean “rank 1 valuation”. DEFINITION. A valuation 1 1 on a field k is a function defined on k with values in the non-negative real numbers satisfying the foIlowing axioms.
The theorem states that f is just I/II in the normalized valuation. [The theory of locally compact topological groups leads to the consideration of the dual (character) group of k+ . It turns out that it is isomorphic to k+. We do not need this fact for class field theory so do not prove it here. For a proof and applications see Tate’s thesis (Chapter XV of this book) or Lang: “Algebraic Numbers” (Addison Wesley), and, for generalizations; Weil: “Adeles and Algebraic Groups” (Princeton lecture notes) and Godement: Bourbaki seminars 171 and 176.
Examples of Valuations The archetypal example of an arch. valuation is the absolute value on the field C of complex numbers. It is essentially the only one: Any field k with an arch. valuation is THEOREM (Gelfand-Tornheim). isomorphic to a subfield of C, the valuation being equivalent to that induced by the absolute valuation on C. We do not prove this as we do not need it. g. E. Artin, “Theory of Algebraic Numbers” (Striker, Gottingen), pp. 45 and 67. The non-arch. valuations are legion. On the rationals Q there is one for every prime p > 0, the p-adic valuation defined by IP”uIvIp = P-’ for a, 24,v E Z, p $ u, p y 0.
Algebraic Number Theory: Proceedings of an Instructional Conference Organized by the London Mathematical Society (A Nato Advanced Study Institute W) by J. W. S. Cassels, A. Frohlich