# 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 norm

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 norm

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 also

## References

- 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 - 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 - Marcus, Daniel A. (1977),
*Number fields*, Universitext, New York: Springer-Verlag, Theorem 22c, ISBN 0-387-90279-1, MR 0457396 - 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