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Local field. In mathematics, a field K is called a (non-Archimedean) local field if it is complete with respect to a topology induced by a discrete valuation v and if its residue field k is finite. [1] Equivalently, a local field is a locally compact topological field with respect to a non-discrete topology. [2]
Local Fields. Textbook. © 1979. Download book PDF. Overview. Authors: Jean-Pierre Serre 0. Collège de France, Paris, France. Part of the book series: Graduate Texts in Mathematics (GTM, volume 67) 176k Accesses. 645 Citations. 6 Altmetric. Softcover Book USD 16.99 USD 74.95. Discount appliedPrice excludes VAT (USA) Compact, lightweight edition.
Hace 6 días · The Hasse principle is one of the chief applications of local field theory. A local field with field characteristic is isomorphic to the field of power series in one variable whose coefficients are in a finite field. A local field of characteristic zero is either the p-adic numbers, or power series in a complex variable.
Theorem 6.13. (i) Let L/K be finite totally ramified, πL ∈ OL a uniformizer. Then the minimal poly-nomial of πL is Eisenstein, OL = OK[πL] and L = K(πL). (ii) Conversely, if f(x) ∈ OK[x] is Eisenstein and α is a root of f, then L = K(α) is a totally ramified extension of K and α is a uniformizer in L.
20 de dic. de 2014 · Such fields are called local, in contrast to global fields (finite extensions of the fields $\mathbf {Q}$ or $k (T)$), and are means for studying the latter. For cohomological properties of Galois extensions of local fields see [1], and also Adèle; Idèle; and Class field theory .
The p-adic numbers, the earliest of local fields, were introduced by Hensel some 70 years ago as a natural tool in algebra number theory. Today the use of this and other local fields pervades much of mathematics, yet these simple and natural concepts, which often provide remarkably easy solutions to complex problems, are not as familiar as they should be.
The theory of local fields, or more generally complete discrete valuation fields, is a widely used tool in algebraic and arithmetic geometry. In particular, such fields are at the heart of the local-to-global principle, the idea that one can study a family of local problems and then deduce global information, typically using the ring of ad`eles.
