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 formula_1 and formula_2 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
formula_3
is the unique group homomorphism that satisfies
formula_4
for all nonzero prime ideals formula_5 of "B", where formula_6 is the prime ideal of "A" lying below formula_5.
Alternatively, for any formula_8 one can equivalently define formula_9 to be the fractional ideal of "A" generated by the set formula_10 of field norms of elements of "B".
For formula_11, one has formula_12, where formula_13.
The ideal norm of a principal ideal is thus compatible with the field norm of an element:
formula_14
Let formula_15 be a Galois extension of number fields with rings of integers formula_16.
Then the preceding applies with formula_17, and for any formula_18 we have
formula_19
which is an element of formula_20.
The notation formula_21 is sometimes shortened to formula_22, an abuse of notation that is compatible with also writing formula_22 for the field norm, as noted above.
In the case formula_24, it is reasonable to use positive rational numbers as the range for formula_25 since formula_26 has trivial ideal class group and unit group formula_27, thus each nonzero fractional ideal of formula_26 is generated by a uniquely determined positive rational number.
Under this convention the relative norm from formula_29 down to formula_24 coincides with the absolute norm defined below.
Absolute norm.
Let formula_29 be a number field with ring of integers formula_32, and formula_33 a nonzero (integral) ideal of formula_32.
The absolute norm of formula_33 is
formula_36
By convention, the norm of the zero ideal is taken to be zero.
If formula_37 is a principal ideal, then
formula_38.
The norm is completely multiplicative: if formula_33 and formula_40 are ideals of formula_32, then
formula_42.
Thus the absolute norm extends uniquely to a group homomorphism
formula_43
defined for all nonzero fractional ideals of formula_32.
The norm of an ideal formula_33 can be used to give an upper bound on the field norm of the smallest nonzero element it contains:
there always exists a nonzero formula_46 for which
formula_47
where
* formula_48 is the discriminant of formula_29 and
* formula_50 is the number of pairs of (non-real) complex embeddings of "L" into formula_51 (the number of complex places of "L").
References.
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