Hasse norm theorem

In number theory, the Hasse norm theorem tells us that if L/K is a cyclic extension of number fields, 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 $\mathbf {N} _{L/K}(l)=k$; 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 L_P; 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 if the extension is abelian but not cyclic.

This is an example of a theorem stating a local-global principle, and is due to Helmut Hasse.