Hasse norm theorem

In number theory, the Hasse norm theorem states 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 in general if the extension is abelian but not cyclic. A counter-example is given by the field ${\mathbf Q}(\sqrt{13},\sqrt{17})/{\mathbf Q}$ 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, and is due to Helmut Hasse.

References

• H. Hasse, "A history of class field theory", in J.W.S. Cassels and A. Frohlich (edd), Algebraic number theory, Academic Press, 1973. Chap.XI.
• G. Janusz, Algebraic number fields, Academic Press, 1973. Theorem V.4.5, p.156