Hasse Norm Theorem
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In
number theory Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...
, the Hasse norm theorem states that if L/K is a
cyclic extension In abstract algebra, an abelian extension is a Galois extension whose Galois group is abelian. When the Galois group is also cyclic, the extension is also called a cyclic extension. Going in the other direction, a Galois extension is called solv ...
of
number field In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a ...
s, then if a nonzero element of K is a local norm everywhere, then it is a global norm. Here to be a global norm means to be an element ''k'' of K such that there is an element ''l'' of L with \mathbf_(l) = k; in other words ''k'' is a relative norm of some element of the extension field L. To be a local norm means that for some prime p of K and some prime P of L lying over K, then ''k'' is a norm from LP; here the "prime" p can be an archimedean valuation, and the theorem is a statement about completions in all valuations, archimedean and non-archimedean. The theorem is no longer true in general if the extension is abelian but not cyclic. Hasse gave the counterexample that 3 is a local norm everywhere for the extension (\sqrt,\sqrt)/ but is not a global norm. Serre and Tate showed that another counterexample is given by the field (\sqrt,\sqrt)/ where every rational square is a local norm everywhere but 5^2 is not a global norm. This is an example of a theorem stating a
local-global principle In mathematics, Helmut Hasse's local–global principle, also known as the Hasse principle, is the idea that one can find an diophantine equation, integer solution to an equation by using the Chinese remainder theorem to piece together solutions mo ...
. The full theorem is due to . The special case when the degree ''n'' of the extension is 2 was proved by , and the special case when ''n'' is prime was proved by Furtwangler in 1902. The Hasse norm theorem can be deduced from the theorem that an element of the
Galois cohomology In mathematics, Galois cohomology is the study of the group cohomology of Galois modules, that is, the application of homological algebra to modules for Galois groups. A Galois group ''G'' associated with a field extension ''L''/''K'' acts in a na ...
group H2(''L''/''K'') is trivial if it is trivial locally everywhere, which is in turn equivalent to the deep theorem that the first cohomology of the
idele class group In abstract algebra, an adelic algebraic group is a semitopological group defined by an algebraic group ''G'' over a number field ''K'', and the adele ring ''A'' = ''A''(''K'') of ''K''. It consists of the points of ''G'' having values in ''A''; th ...
vanishes. This is true for all finite Galois extensions of number fields, not just cyclic ones. For cyclic extensions the group H2(''L''/''K'') is isomorphic to the Tate cohomology group H0(''L''/''K'') which describes which elements are norms, so for cyclic extensions it becomes Hasse's theorem that an element is a norm if it is a local norm everywhere.


See also

* Grunwald–Wang theorem, about when an element that is a power everywhere locally is a power.


References

* * H. Hasse, "A history of class field theory", in J.W.S. Cassels and
A. Frohlich A is the first letter of the Latin and English alphabet. A may also refer to: Science and technology Quantities and units * ''a'', a measure for the attraction between particles in the Van der Waals equation * A value, ''A'' value, a measu ...
(edd), ''Algebraic number theory'',
Academic Press Academic Press (AP) is an academic book publisher founded in 1941. It launched a British division in the 1950s. Academic Press was acquired by Harcourt, Brace & World in 1969. Reed Elsevier said in 2000 it would buy Harcourt, a deal complete ...
, 1973. Chap.XI. * G. Janusz, ''Algebraic number fields'', Academic Press, 1973. Theorem V.4.5, p. 156 *{{Citation , last1=Hilbert , first1=David , author1-link=David Hilbert , title=Die Theorie der algebraischen Zahlkörper , url=http://resolver.sub.uni-goettingen.de/purl?GDZPPN002115344 , language=German , year=1897 , journal=Jahresbericht der Deutschen Mathematiker-Vereinigung , issn=0012-0456 , volume=4 , pages=175–546 Class field theory Theorems in algebraic number theory