An alternative description of K^*/Nm(L^*) - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T13:14:11Z http://mathoverflow.net/feeds/question/40221 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/40221/an-alternative-description-of-k-nml An alternative description of K^*/Nm(L^*) Evgeny Shinder 2010-09-27T21:19:52Z 2010-09-28T15:11:44Z <p>Is there a nice explicit description for the group $K^*/Nm_{L/K}(L^*)$ for a finite field extension $L/K$? </p> <p>What if for example, $L$ is obtained from $K$ by ajoining an n-th root of some $\alpha \in K$ (and assuming that $K$ contains the n-th root of unity)? I don't see a nice answer even for the case $n=2$.</p> <p>Thank you. </p> <p>EDIT: Thanks for the answers! Is that correct that in the case of a cyclic extension this group is isomorphic to $Br(L/K)$, since both these groups are identified with $H^2(Gal(L/K), L^*)$?</p> http://mathoverflow.net/questions/40221/an-alternative-description-of-k-nml/40248#40248 Answer by Alex Bartel for An alternative description of K^*/Nm(L^*) Alex Bartel 2010-09-28T01:45:25Z 2010-09-28T02:12:32Z <p>It really depends on what kinds of field $K$ and $L$ are. For example, if they are finite fields, then the quotient vanishes, since the norm is always surjective.</p> <p>If $K$ is a local field with finite residue field and $L/K$ is abelian, then class field theory says that $K^\times/{\rm Norm}_{L/K}(L^\times)$ is isomorphic to the Galois group of $L/K$ (more generally, if $L/K$ is an arbitrary finite extension of local fields, then the quotient is isomorphic to the Galois group of the maximal abelian subextensions of $K$). This isomorphism is reasonably explicit and is described in any good exposition of class field theory. If $K$ is perfect, but not necessarily finite, or a local field with perfect residue field, then one can still say something reasonably explicit (see e.g. Serre's book "Corps Locaux").</p> <p>If $K$ is a global field, then the situation is more complicated and less explicit, because the "right" object to look at, from the point of view of class field theory, is not the norm quotient you have written down, but the same with the multiplicative groups of the fields replaced by idèle class groups.</p> <p>You need to tell us more about the fields in question to get a better answer.</p> <p>Edit: to slightly generalise what I have said above, you can replace "finite field" by "<a href="http://en.wikipedia.org/wiki/Quasi-finite_field" rel="nofollow">quasi-finite field</a>" throughout, and the statements will still hold.</p> http://mathoverflow.net/questions/40221/an-alternative-description-of-k-nml/40261#40261 Answer by Franz Lemmermeyer for An alternative description of K^*/Nm(L^*) Franz Lemmermeyer 2010-09-28T04:20:17Z 2010-09-28T04:20:17Z <p>In addition to Alex's answer I would like to point out that studying the group $K^\times/N(L^\times)$ is a perfectly fine goal (related of course to the validity of the Hasse norm principle in extensions of number fields), which should not simply be dismissed as a "wrong question". A good place to start is the work by Leonid Stern, the most recent article being <em>On the norm groups of Galois $2\frak n$-extensions of algebraic number fields</em>, J. Number Theory 129, No. 5, 1191-1204 (2009).</p>