Absolute 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 ${\displaystyle {\mathcal {I}_{A}$ and ${\displaystyle {\mathcal {I}_{B}$ 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

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

is the unique group homomorphism that satisfies

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

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

Alternatively, for any ${\displaystyle {\mathfrak {b}\in {\mathcal {I}_{B}$ one can equivalently define ${\displaystyle N_{B/A}({\mathfrak {b})}$ to be the fractional ideal of A generated by the set ${\displaystyle \{N_{L/K}(x)|x\in {\mathfrak {b}\}$ of field norms of elements of B.^[1]

For ${\displaystyle {\mathfrak {a}\in {\mathcal {I}_{A}$, one has ${\displaystyle N_{B/A}({\mathfrak {a}B)={\mathfrak {a}^{n}$, where ${\displaystyle n=[L:K]}$.

The ideal norm of a principal ideal is thus compatible with the field norm of an element:

${\displaystyle N_{B/A}(xB)=N_{L/K}(x)A.}$^[2]

Let ${\displaystyle L/K}$ be a Galois extension of number fields with rings of integers ${\displaystyle {\mathcal {O}_{K}\subset {\mathcal {O}_{L}$.

Then the preceding applies with ${\displaystyle A={\mathcal {O}_{K},B={\mathcal {O}_{L}$, and for any ${\displaystyle {\mathfrak {b}\in {\mathcal {I}_{\mathcal {O}_{L}$ we have

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

which is an element of ${\displaystyle {\mathcal {I}_{\mathcal {O}_{K}$.

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

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

Absolute norm

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

The absolute norm of ${\displaystyle {\mathfrak {a}$ is

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

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

If ${\displaystyle {\mathfrak {a}=(a)}$ is a principal ideal, then

${\displaystyle N({\mathfrak {a})=\left|N_{L/\mathbb {Q} }(a)\right|}$.^[3]

The norm is completely multiplicative: if ${\displaystyle {\mathfrak {a}$ and ${\displaystyle {\mathfrak {b}$ are ideals of ${\displaystyle {\mathcal {O}_{L}$, then

${\displaystyle N({\mathfrak {a}\cdot {\mathfrak {b})=N({\mathfrak {a})N({\mathfrak {b})}$.^[3]

Thus the absolute norm extends uniquely to a group homomorphism

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

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

The norm of an ideal ${\displaystyle {\mathfrak {a}$ can be used to give an upper bound on the field norm of the smallest nonzero element it contains:

there always exists a nonzero ${\displaystyle a\in {\mathfrak {a}$ for which

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

where

• ${\displaystyle \Delta _{L}$ is the discriminant of ${\displaystyle L}$ and
• ${\displaystyle s}$ is the number of pairs of (non-real) complex embeddings of L into ${\displaystyle \mathbb {C} }$ (the number of complex places of L).^[4]

See also

• Field norm
• Dedekind zeta function

References