Norm of an ideal

The norm of an ideal is defined in algebraic number theory. Let $Ksubset\; L$ be two number fields with rings of integers $O\_Ksubset\; O\_L$. Suppose that the extension $L/K$ is a Galois extension with

:$G=\; extstyle\{Gal\}(L/K).$

The norm of an ideal $I$ of $O\_L$ is defined as follows

:$N\_K^L(I)=O\_K\; capprod\_\{sigma\; in\; G\}^\{\}\; sigma\; (I)$

which is an ideal of $O\_K$. The norm of a principal ideal generated by "α" is the ideal generated by the field norm of "α".

The norm map is defined from the set of ideals of $O\_L$. to the set of ideals of $O\_K$. It is reasonable to use integers as the range for the norm map

:$N\_mathbb\{Q\}^L(I)$

since Z is a principal ideal domain. This idea doesn't work in general since class group is usually non-trivial.

Alternate Formulation

Let $L$ be a number field with ring of integers $O\_L$, and $alpha$ a nonzero ideal of $O\_L$. Then the norm of $alpha$ is defined to be :$N(alpha)\; =left\; [\; O\_L:\; alpha\; ight\; ]\; =|O\_L/alpha|$.By convention, the norm of the zero ideal is taken to be zero.

If $alpha$ is a principal ideal with $alpha=(a)$, then $N(alpha)=|N(a)|$.

The norm is also completely multiplicative in that if $alpha$ and are ideals of $O\_L$, then .

The norm of an ideal $alpha$ can be used to bound the norm of some nonzero element $xin\; alpha$ by the inequality:$|N(x)|leq\; left\; (\; frac\{2\}\{pi\}\; ight\; )\; ^\; \{r\_2\}\; sqrtN(alpha)$where $Delta\_L$ is the discriminant of $L$ and $r\_2$ is the number of pairs of complex embeddings of $L$ into $mathbb\{C\}$.

ee also

*Dedekind zeta function

References

* Daniel A. Marcus, "Number Fields", third edition, Springer-Verlag, 1977