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.
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 (Serre 1979), 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 .
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.
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 .
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
- Janusz, Proposition I.8.2
- Serre, 1.5, Proposition 14.
- Marcus, Theorem 22c, pp. 65-66.
- Marcus, Theorem 22a, pp. 65-66
- Neukirch, Lemma 6.2
- Janusz, Gerald J. (1996), Algebraic number fields, Graduate Studies in Mathematics 7 (second ed.), Providence, Rhode Island: American Mathematical Society, pp. x+276, ISBN 0-8218-0429-4, MR 1362545 (96j:11137)
- Marcus, Daniel A. (1977), Number fields, Universitext, New York: Springer-Verlag, pp. viii+279, ISBN 0-387-90279-1, MR 0457396 (56 #15601)
- Jürgen Neukirch (1999), Algebraic number theory, Berlin: Springer-Verlag, pp. xviii+571, ISBN 3-540-65399-6, MR 1697859 (2000m:11104)
- Serre, Jean-Pierre (1979), Local fields, Graduate Texts in Mathematics 67, Translated from the French by Marvin Jay Greenberg, New York: Springer-Verlag, pp. viii+241, ISBN 0-387-90424-7, MR 554237 (82e:12016)