= Hasse norm theorem
{c}
{title2=$K^\times\cap N_{L/K}J_L=N_{L/K}L^\times\quad(L/K\text{ cyclic})$}
{wiki}
For a cyclic extension of <number fields>, an element of the base field is a global <field norm> exactly when it is a norm at every finite and infinite completion. The <splitting criterion for a cyclic algebra> encodes the norm obstruction as a <Brauer group> class. The <Albert-Brauer-Hasse-Noether theorem> makes a class that splits at every completion split globally, giving the result. Cyclicity is essential; arbitrary noncyclic extensions can have additional local-global norm obstructions.
Back to article page
