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.{{citation

|last=Janusz

|first=Gerald J.

|title=Algebraic number fields

|edition=second

|series=Graduate Studies in Mathematics

|volume=7

|publisher=American Mathematical Society

|place=Providence, Rhode Island

|date=1996

|isbn=0-8218-0429-4

|mr=1362545

|at=Proposition I.8.2

}}

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:

: $N_{B/A}(xB) = N_{L/K}(x)A.$ {{citation

|last=Serre

|first=Jean-Pierre

|title=Local Fields

|authorlink1= Jean-Pierre Serre

|series=Graduate Texts in Mathematics

|volume=67

|translator-link=Marvin Greenberg

|translator-first=Marvin Jay

|translator-last1=Greenberg

|publisher=Springer-Verlag

|place=New York

|date=1979

|isbn=0-387-90424-7

|mr=554237

|at=1.5, Proposition 14

}}

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

: $N(\mathfrak a)=\left|N_{L/\mathbb{Q}}(a)\right|$ .{{citation

|last=Marcus

|first=Daniel A.

|title=Number fields

|series=Universitext

|publisher=Springer-Verlag

|place=New York

|date=1977

|isbn=0-387-90279-1

|mr=0457396

|at=Theorem 22c

}}

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 {{math|L}} into $\mathbb{C}$ (the number of complex places of {{math|L}}).{{citation

|first=Jürgen

|last=Neukirch

|title=Algebraic number theory

|series=Grundlehren der mathematischen Wissenschaften

|publisher=Springer-Verlag

|place=Berlin

|date=1999

|volume=322

|isbn=3-540-65399-6

|at=Lemma 6.2

|mr=1697859

|doi=10.1007/978-3-662-03983-0

}}

References

Category:Algebraic number theory

Category:Commutative algebra

Category:Ideals (ring theory)

Ideal norm

Relative norm

Absolute norm

See also

References