Ideal norm

In commutative algebra, the norm of an ideal is a generalization of a norm of an element in the field extension. It is particularly important in number theory since it measures the size of an ideal of a complicated number ring in terms of an ideal in a less complicated ring. When the less complicated number ring is taken to be the ring of integers, Z, then the norm of a nonzero ideal I of a number ring R is simply the size of the finite quotient ring R/I.

Relative normEdit

Let A be a Dedekind domain with field of fractions K and integral closure of B in a finite separable extension L of K. (this implies that B is also a Dedekind domain.) Let   and   be the ideal groups of A and B, respectively (i.e., the sets of nonzero fractional ideals.) Following the technique developed by Jean-Pierre Serre, the norm map

 

is the unique group homomorphism that satisfies

 

for all nonzero prime ideals   of B, where   is the prime ideal of A lying below  .

Alternatively, for any   one can equivalently define   to be the fractional ideal of A generated by the set   of field norms of elements of B.[1]

For  , one has  , where  . The ideal norm of a principal ideal is thus compatible with the field norm of an element:  [2]

Let   be a Galois extension of number fields with rings of integers  . Then the preceding applies with  , and for any   we have

 

which is an element of  . The notation   is sometimes shortened to  , an abuse of notation that is compatible with also writing   for the field norm, as noted above.

In the case  , it is reasonable to use positive rational numbers as the range for   since   has trivial ideal class group and unit group  , thus each nonzero fractional ideal of   is generated by a uniquely determined positive rational number. Under this convention the relative norm from   down to   coincides with the absolute norm defined below.

Absolute normEdit

Let   be a number field with ring of integers  , and   a nonzero (integral) ideal of  . The absolute norm of   is

 

By convention, the norm of the zero ideal is taken to be zero.

If   is a principal ideal, then  .[3]

The norm is completely multiplicative: if   and   are ideals of  , then  .[3] Thus the absolute norm extends uniquely to a group homomorphism

 

defined for all nonzero fractional ideals of  .

The norm of an ideal   can be used to give an upper bound on the field norm of the smallest nonzero element it contains: there always exists a nonzero   for which

 

where   is the discriminant of   and   is the number of pairs of (non-real) complex embeddings of L into   (the number of complex places of L).[4]

See alsoEdit

ReferencesEdit

  1. ^ Janusz, Gerald J. (1996), Algebraic number fields, Graduate Studies in Mathematics, 7 (second ed.), Providence, Rhode Island: American Mathematical Society, Proposition I.8.2, ISBN 0-8218-0429-4, MR 1362545
  2. ^ Serre, Jean-Pierre (1979), Local Fields, Graduate Texts in Mathematics, 67, translated by Greenberg, Marvin Jay, New York: Springer-Verlag, 1.5, Proposition 14, ISBN 0-387-90424-7, MR 0554237
  3. ^ a b Marcus, Daniel A. (1977), Number fields, Universitext, New York: Springer-Verlag, Theorem 22c, ISBN 0-387-90279-1, MR 0457396
  4. ^ Neukirch, Jürgen (1999), Algebraic number theory, Berlin: Springer-Verlag, Lemma 6.2, doi:10.1007/978-3-662-03983-0, ISBN 3-540-65399-6, MR 1697859