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
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