Theorem

Hasse Norm Theorem

Number Theory

The Hasse norm theorem is a result in number theory stating that for a cyclic extension of number fields L over K, a nonzero element of K that is a local norm at every completion of K is in fact a global norm, meaning it is the relative norm of some element of L. The theorem is an example of a local-global principle, and it no longer holds in general when the extension is abelian but not cyclic; Hasse himself gave a counterexample, and Serre and Tate later gave another. The full theorem is due to Helmut Hasse in 1931, building on earlier special cases proved by Hilbert in 1897 and by Furtwangler in 1902. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/

Facts

Statement

If L/K is a cyclic extension of number fields and a nonzero element of K is a local norm everywhere, then it is a global norm. 1

Proof Year

1931 1

Classification

Statement Form

Characterization Theorem 1

Sources

1. Hasse norm theorem (Wikipedia)

Introduction, first sentence
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
Introduction, fifth paragraph
Hasse (1931)

Switch Atlas

The Inner World

The Civil World

The Natural World

Hasse Norm Theorem