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 ; 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. A counter-example is given by the field where every rational square is a local norm everywhere but is not a global norm.
This is an example of a theorem stating a local-global principle, and is due to Helmut Hasse.
Famous quotes containing the words norm and/or theorem:
“To be told that our child’s behavior is “normal” offers little solace when our feelings are badly hurt, or when we worry that his actions are harmful at the moment or may be injurious to his future. It does not help me as a parent nor lessen my worries when my child drives carelessly, even dangerously, if I am told that this is “normal” behavior for children of his age. I’d much prefer him to deviate from the norm and be a cautious driver!”
—Bruno Bettelheim (20th century)
“To insure the adoration of a theorem for any length of time, faith is not enough, a police force is needed as well.”
—Albert Camus (1913–1960)