; 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
Serre and Tate showed that another counterexample is given by the field
The special case when the degree n of the extension is 2 was proved by Hilbert (1897), and the special case when n is prime was proved by Furtwangler in 1902.
[citation needed] The Hasse norm theorem can be deduced from the theorem that an element of the Galois cohomology 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 vanishes.
This is true for all finite Galois extensions of number fields, not just cyclic ones.