Relative Norm
Let A be a Dedekind domain with the field of fractions K and B be the integral closure of A in a finite separable extension L of K. (In particular, B is Dedekind then.) Let and be the ideal groups of A and B, respectively (i.e., the sets of fractional ideals.) Following (Serre 1979), the norm map
is a homomorphism given by
If are local fields, is defined to be a fractional ideal generated by the set This definition is equivalent to the above and is given in (Iwasawa 1986).
For, one has where . The definition is thus also compatible with norm of an element:
Let be a finite Galois extension of number fields with rings of integer . Then the preceding applies with and one has
which is an ideal of . The norm of a principal ideal generated by α is the ideal generated by the field norm of α.
The norm map is defined from the set of ideals of to the set of ideals of . It is reasonable to use integers as the range for since Z has trivial ideal class group. This idea does not work in general since the class group may not be trivial.
Read more about this topic: Ideal Norm
Famous quotes containing the words relative and/or norm:
“The ungentlemanly expressions and gasconading conduct of yours relative to me yesterday was in true character of yourself and unmask you to the world and plainly show that they were ebullitions of a base mind ... and flow from a source devoid of every refined sentiment or delicate sensations.”
—Andrew Jackson (1767–1845)
“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)