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 (Serre 1979), 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 ${\mathfrak {q}}$.

If $L,K$ are local fields with discrete valuation rings $B,A$, then $N_{B/A}({\mathfrak {b}})$ is defined to be a fractional ideal generated by the set $\{N_{L/K}(x)|x\in {\mathfrak {b}}\}$, where $N_{L/K}\colon L\to K$ is the field norm. This definition is equivalent to the above and is given in (Iwasawa 1986).

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

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

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 $N_{K/K}$ 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 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. In the case of integral ideals, this coincides with the absolute norm defined below.

Absolute norm

Let $L$ be a number field with ring of integers ${\mathcal {O}}_{L}$, and $\alpha$ a nonzero ideal of ${\mathcal {O}}_{L}$. Then the norm of $\alpha$ is defined to be

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

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

If $\alpha$ is a principal ideal with $\alpha =(a)$, then $N(\alpha )=|N(a)|$. For proof, cf. Marcus, theorem 22c, pp65ff.

The norm is also completely multiplicative in that if $\alpha$ and $\beta$ are ideals of ${\mathcal {O}}_{L}$, then $N(\alpha \cdot \beta )=N(\alpha )N(\beta )$. For proof, cf. Marcus, theorem 22a, pp65ff.

The norm of an ideal $\alpha$ can be used to bound the norm of some nonzero element $x\in \alpha$ by the inequality

$|N(x)|\leq \left({\frac {2}{\pi }}\right)^{r_{2}}{\sqrt {|\Delta _{L}|}}N(\alpha )$

where $\Delta _{L}$ is the discriminant of $L$ and $r_{2}$ is the number of pairs of complex embeddings of $L$ into $\mathbf {C}$.

See also

• Dedekind zeta function

References

• Iwasawa, Kenkichi (1986), Local class field theory, Oxford Mathematical Monographs, New York: The Clarendon Press Oxford University Press, pp. viii+155, ISBN 0-19-504030-9, MR 863740 (88b:11080) {{citation}}: Check |mr= value (help)
• Marcus, Daniel A. (1977), Number fields, Universitext, New York: Springer-Verlag, pp. viii+279, ISBN 0-387-90279-1, MR 0457396 (56 #15601) {{citation}}: Check |mr= value (help)
• Serre, Jean-Pierre (1979), Local fields, Graduate Texts in Mathematics, vol. 67, Translated from the French by Marvin Jay Greenberg, New York: Springer-Verlag, pp. viii+241, ISBN 0-387-90424-7, MR 554237 (82e:12016) {{citation}}: Check |mr= value (help)