Ideal norm

 ( Algebraic Number Theory ) 

In commutative algebra, the norm of an ideal is a generalization of a field 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.

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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 $\backslash mathcal\{I\}\_A$ and $\backslash mathcal\{I\}\_B$ be the of A and B, respectively (i.e., the sets of nonzero .) Following the technique developed by Jean-Pierre Serre, the norm map

$N\_\{B/A\}\backslash colon\; \backslash mathcal\{I\}\_B\; \backslash to\; \backslash mathcal\{I\}\_A$

is the unique group homomorphism that satisfies

$N\_\{B/A\}(\backslash mathfrak\; q)\; =\; \backslash mathfrak\{p\}^\{B/\backslash mathfrak\}$

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

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

For $\backslash mathfrak\; a\; \backslash in\; \backslash mathcal\{I\}\_A$, one has $N\_\{B/A\}(\backslash mathfrak\; a\; B)\; =\; \backslash mathfrak\; a^n$, where $n\; =\; L$.

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 with rings of integers $\backslash mathcal\{O\}\_K\backslash subset\; \backslash mathcal\{O\}\_L$.

Then the preceding applies with $A\; =\; \backslash mathcal\{O\}\_K,\; B\; =\; \backslash mathcal\{O\}\_L$, and for any $\backslash mathfrak\; b\backslash in\backslash mathcal\{I\}\_\{\backslash mathcal\{O\}\_L\}$ we have

$N\_\{\backslash mathcal\{O\}\_L/\backslash mathcal\{O\}\_K\}(\backslash mathfrak\; b)=\; K\; \backslash cap\backslash prod\_\{\backslash sigma\; \backslash in\; \backslash operatorname\{Gal\}(L/K)\}\; \backslash sigma\; (\backslash mathfrak\; b),$

which is an element of $\backslash mathcal\{I\}\_\{\backslash mathcal\{O\}\_K\}$.

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

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Absolute norm Let $L$ be a number field with ring of integers $\backslash mathcal\{O\}\_L$, and $\backslash mathfrak\; a$ a nonzero (integral) ideal of $\backslash mathcal\{O\}\_L$.

The absolute norm of $\backslash mathfrak\; a$ is

$N(\backslash mathfrak\; a)\; :=\backslash left\; =\backslash left|\backslash mathcal\{O\}\_L/\backslash mathfrak\; a\backslash right|.\backslash ,$

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

If $\backslash mathfrak\; a=(a)$ is a principal ideal, then

$N(\backslash mathfrak\; a)=\backslash left|N\_\{L/\backslash mathbb\{Q\}\}(a)\backslash right|$.

The norm is completely multiplicative: if $\backslash mathfrak\; a$ and $\backslash mathfrak\; b$ are ideals of $\backslash mathcal\{O\}\_L$, then

$N(\backslash mathfrak\; a\backslash cdot\backslash mathfrak\; b)=N(\backslash mathfrak\; a)N(\backslash mathfrak\; b)$.

Thus the absolute norm extends uniquely to a group homomorphism

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

defined for all nonzero of $\backslash mathcal\{O\}\_L$.

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

there always exists a nonzero $a\backslash in\backslash mathfrak\; a$ for which

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

where

* $\backslash Delta\_L$ is the discriminant of $L$ and

* $s$ is the number of pairs of (non-real) complex of into $\backslash mathbb\{C\}$ (the number of complex places of ).

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See also

• Field norm
• Dedekind zeta function

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