## Interactions Between Ring Theory and Representations of by Freddy Van Oystaeyen, Manolo Saorin

By Freddy Van Oystaeyen, Manolo Saorin

This paintings is predicated on a suite of lectures and invited papers provided at a gathering in Murcia, Spain, equipped by means of the ecu Commission's education and Mobility of Researchers (TMR) Programme. It includes info at the constitution of illustration concept of teams and algebras and on common ring theoretic equipment with regards to the idea.

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6. Totally Ramified Extensions The notation here, and in $ 7 to 9, is that introduced in 5 5. v. ring, K is complete and L is a finite, separable extension field of K. A polynomial g(X) in K[X] is separable if (g(X), g’(X)) = 1. +b,X+bO, (1) t See E. Noether, Normalbasis bei Kiirpern ohne hiihere Verzweigung, Creole 1931. $ See R. S. Swan, Induced Representations and Projective Modules, Ann. of Math. 4. LOCAL 23 FIELDS with trg(bJ 2 1 for i = 1,. . , m- 1, and u&J = 1. (The condition of separability on either L or E(x) is not really necessary for the following theorem.

The composite field of non-ramified extensions L and L in a given separable closure of K is non-ramified. The union K, of all non-ramified extensions L of K in a given separable closure of K is called the maximal non-ramified extension of K. COROLLARY 2. Every finite extension of K in K,,, is non-ramiJied. The Galois group I’(K,,JK) is isomorphic (as a topological group) with the Galois group I’(P/k) of the separable closure 7i” of k. Application (see Chapters III and V). We suppose now that k is a finite field of characteristicp with q = pm elements.

We define l as a metric space to be the completion of k as a metric space with respect to I I. Since the field operations + , x and inverse are continuous on k they are well-defined on ff. D. COROLLARY 1. I I is non-arch. on & if and only if it is so on k. If that is so, the set of values taken by I I on k and It are the same. Proof. Use second lemma of 5 2. , the functional inequality (a fundamental IS+YI I max