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.^[1]

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

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

The norm is completely multiplicative: if ${\mathfrak {a}}$ and ${\mathfrak b}$ are ideals of ${\mathcal {O}}_{L}$ , then $N({\mathfrak a}\cdot {\mathfrak b})=N({\mathfrak a})N({\mathfrak b})$ .^[3] 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 $L$ into $\mathbb {C}$ (the number of complex places of $L$ ).^[4]

References

↑ Janusz, Gerald J. (1996), Algebraic number fields, Graduate Studies in Mathematics, 7 (second ed.), Providence, Rhode Island: American Mathematical Society, Proposition I.8.2, ISBN 0-8218-0429-4, MR 1362545 (96j:11137)
↑ Serre, Jean-Pierre (1979), Local fields, Graduate Texts in Mathematics, 67, Translated from the French by Marvin Jay Greenberg, New York: Springer-Verlag, 1.5, Proposition 14, ISBN 0-387-90424-7, MR 554237 (82e:12016)
1 2 Marcus, Daniel A. (1977), Number fields, Universitext, New York: Springer-Verlag, Theorem 22c, ISBN 0-387-90279-1, MR 0457396 (56 #15601)
↑ Neukirch, Jürgen (1999), Algebraic number theory, Berlin: Springer-Verlag, Lemma 6.2, ISBN 3-540-65399-6, MR 1697859 (2000m:11104)

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Ideal norm

Relative norm

Absolute norm

See also

References