Atle Selberg Collected Papers by Atle Selberg PDF

By Atle Selberg

ISBN-10: 0387183892

ISBN-13: 9780387183893

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Hence they give us all automorphisms of the residue class field, as was to be shown. Corollary 1. Let A be a ring integrally closecl in its quotient fielcl K. Let L be a finite Galois extension of K, ancl B the integral closure of A in L. Let lJ be a maxima[ ideal of A. Let rp: A ---7 A/p be the canonical homomorphism, ancl let 1/1 1 , 1/; 2 be two homomorphisms of B extencling rp in a given algebraic closure of A/p. Then there exists an automorphism u of L over K such that Proof. The kernels of 1/; 1 , 1/; 2 are prime ideals of B which are conjugate by Proposition 11.

Then we have the ordinary absolute value. For each prime number p we have the p-adic valuation Vp = liP, defined by the formula where ris an integer, and m, n are integers ~ O and not divisible by p. Let o be a discrete valuation ring with maximal ideal m, generated by an element 1r. Every non-zero element a of the quotient field K of o can be written in the forma = 1rru, where ris an integer and u is a unit in o. We call r the order of a. Let c be a positive real number, O < c < 1. If we define lai= cr, then we get an absolute value on K (trivial verification), which is in fact a valuation.

This proves what we wanted. Remark. In all the above propositions, we could assume lJ prime instead of maximal. In that case, one has to localize at lJ tobe able to apply our proofs. In the application to number fields, this is unnecessary, since every prime is maximal. In the above discussions, the kernel of the map is called the inertia group T'l3 of '13. It consists of those automorphisms of G'l3 which induce the trivial automorphism on the residue class field. Its fixed field is called the inertia field, and is denoted by Lt.

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Atle Selberg Collected Papers by Atle Selberg


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