# 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 ${\mathcal {I}}_{A}$  and ${\mathcal {I}}_{B}$  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

$N_{B/A}\colon {\mathcal {I}}_{B}\to {\mathcal {I}}_{A}$

is the unique group homomorphism that satisfies

$N_{B/A}({\mathfrak {q}})={\mathfrak {p}}^{[B/{\mathfrak {q}}:A/{\mathfrak {p}}]}$

for all nonzero prime ideals ${\mathfrak {q}}$  of B, where ${\mathfrak {p}}={\mathfrak {q}}\cap A$  is the prime ideal of A lying below ${\mathfrak {q}}$ .

Alternatively, for any ${\mathfrak {b}}\in {\mathcal {I}}_{B}$  one can equivalently define $N_{B/A}({\mathfrak {b}})$  to be the fractional ideal of A generated by the set $\{N_{L/K}(x)|x\in {\mathfrak {b}}\}$  of field norms of elements of B.

For ${\mathfrak {a}}\in {\mathcal {I}}_{A}$ , one has $N_{B/A}({\mathfrak {a}}B)={\mathfrak {a}}^{n}$ , where $n=[L:K]$ . The ideal norm of a principal ideal is thus compatible with the field norm of an element: $N_{B/A}(xB)=N_{L/K}(x)A.$ 

Let $L/K$  be a Galois extension of number fields with rings of integers ${\mathcal {O}}_{K}\subset {\mathcal {O}}_{L}$ . Then the preceding applies with $A={\mathcal {O}}_{K},B={\mathcal {O}}_{L}$ , and for any ${\mathfrak {b}}\in {\mathcal {I}}_{{\mathcal {O}}_{L}}$  we have

$N_{{\mathcal {O}}_{L}/{\mathcal {O}}_{K}}({\mathfrak {b}})=K\cap \prod _{\sigma \in \operatorname {Gal} (L/K)}\sigma ({\mathfrak {b}}),$

which is an element of ${\mathcal {I}}_{{\mathcal {O}}_{K}}$ . The notation $N_{{\mathcal {O}}_{L}/{\mathcal {O}}_{K}}$  is sometimes shortened to $N_{L/K}$ , an abuse of notation that is compatible with also writing $N_{L/K}$  for the field norm, as noted above.

In the case $K=\mathbb {Q}$ , it is reasonable to use positive rational numbers as the range for $N_{{\mathcal {O}}_{L}/\mathbb {Z} }\,$  since $\mathbb {Z}$  has trivial ideal class group and unit group $\{\pm 1\}$ , thus each nonzero fractional ideal of $\mathbb {Z}$  is generated by a uniquely determined positive rational number. Under this convention the relative norm from $L$  down to $K=\mathbb {Q}$  coincides with the absolute norm defined below.

## Absolute norm

Let $L$  be a number field with ring of integers ${\mathcal {O}}_{L}$ , and ${\mathfrak {a}}$  a nonzero (integral) ideal of ${\mathcal {O}}_{L}$ . The absolute norm of ${\mathfrak {a}}$  is

$N({\mathfrak {a}}):=\left[{\mathcal {O}}_{L}:{\mathfrak {a}}\right]=\left|{\mathcal {O}}_{L}/{\mathfrak {a}}\right|.\,$

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

If ${\mathfrak {a}}=(a)$  is a principal ideal, then $N({\mathfrak {a}})=\left|N_{L/\mathbb {Q} }(a)\right|$ .

The norm is completely multiplicative: if ${\mathfrak {a}}$  and ${\mathfrak {b}}$  are ideals of ${\mathcal {O}}_{L}$ , then $N({\mathfrak {a}}\cdot {\mathfrak {b}})=N({\mathfrak {a}})N({\mathfrak {b}})$ . Thus the absolute norm extends uniquely to a group homomorphism

$N\colon {\mathcal {I}}_{{\mathcal {O}}_{L}}\to \mathbb {Q} _{>0}^{\times },$

defined for all nonzero fractional ideals of ${\mathcal {O}}_{L}$ .

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

$\left|N_{L/\mathbb {Q} }(a)\right|\leq \left({\frac {2}{\pi }}\right)^{s}{\sqrt {\left|\Delta _{L}\right|}}N({\mathfrak {a}}),$

where $\Delta _{L}$  is the discriminant of $L$  and $s$  is the number of pairs of (non-real) complex embeddings of L into $\mathbb {C}$  (the number of complex places of L).