Английская Википедия: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 <math>\mathcal{I}_A</math> and <math>\mathcal{I}_B</math> 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
- <math>N_{B/A}\colon \mathcal{I}_B \to \mathcal{I}_A</math>
is the unique group homomorphism that satisfies
- <math>N_{B/A}(\mathfrak q) = \mathfrak{p}^{[B/\mathfrak q : A/\mathfrak p]}</math>
for all nonzero prime ideals <math>\mathfrak q</math> of B, where <math>\mathfrak p = \mathfrak q\cap A</math> is the prime ideal of A lying below <math>\mathfrak q</math>.
Alternatively, for any <math>\mathfrak b\in\mathcal{I}_B</math> one can equivalently define <math>N_{B/A}(\mathfrak{b})</math> to be the fractional ideal of A generated by the set <math>\{ N_{L/K}(x) | x \in \mathfrak{b} \}</math> of field norms of elements of B.[1]
For <math>\mathfrak a \in \mathcal{I}_A</math>, one has <math>N_{B/A}(\mathfrak a B) = \mathfrak a^n</math>, where <math>n = [L : K]</math>.
The ideal norm of a principal ideal is thus compatible with the field norm of an element:
- <math>N_{B/A}(xB) = N_{L/K}(x)A.</math>[2]
Let <math>L/K</math> be a Galois extension of number fields with rings of integers <math>\mathcal{O}_K\subset \mathcal{O}_L</math>.
Then the preceding applies with <math>A = \mathcal{O}_K, B = \mathcal{O}_L</math>, and for any <math>\mathfrak b\in\mathcal{I}_{\mathcal{O}_L}</math> we have
- <math>N_{\mathcal{O}_L/\mathcal{O}_K}(\mathfrak b)= K \cap\prod_{\sigma \in \operatorname{Gal}(L/K)} \sigma (\mathfrak b),</math>
which is an element of <math>\mathcal{I}_{\mathcal{O}_K}</math>.
The notation <math>N_{\mathcal{O}_L/\mathcal{O}_K}</math> is sometimes shortened to <math>N_{L/K}</math>, an abuse of notation that is compatible with also writing <math>N_{L/K}</math> for the field norm, as noted above.
In the case <math>K=\mathbb{Q}</math>, it is reasonable to use positive rational numbers as the range for <math>N_{\mathcal{O}_L/\mathbb{Z}}\,</math> since <math>\mathbb{Z}</math> has trivial ideal class group and unit group <math>\{\pm 1\}</math>, thus each nonzero fractional ideal of <math>\mathbb{Z}</math> is generated by a uniquely determined positive rational number.
Under this convention the relative norm from <math>L</math> down to <math>K=\mathbb{Q}</math> coincides with the absolute norm defined below.
Absolute norm
Let <math>L</math> be a number field with ring of integers <math>\mathcal{O}_L</math>, and <math>\mathfrak a</math> a nonzero (integral) ideal of <math>\mathcal{O}_L</math>.
The absolute norm of <math>\mathfrak a</math> is
- <math>N(\mathfrak a) :=\left [ \mathcal{O}_L: \mathfrak a\right ]=\left|\mathcal{O}_L/\mathfrak a\right|.\,</math>
By convention, the norm of the zero ideal is taken to be zero.
If <math>\mathfrak a=(a)</math> is a principal ideal, then
- <math>N(\mathfrak a)=\left|N_{L/\mathbb{Q}}(a)\right|</math>.[3]
The norm is completely multiplicative: if <math>\mathfrak a</math> and <math>\mathfrak b</math> are ideals of <math>\mathcal{O}_L</math>, then
- <math>N(\mathfrak a\cdot\mathfrak b)=N(\mathfrak a)N(\mathfrak b)</math>.[3]
Thus the absolute norm extends uniquely to a group homomorphism
- <math>N\colon\mathcal{I}_{\mathcal{O}_L}\to\mathbb{Q}_{>0}^\times,</math>
defined for all nonzero fractional ideals of <math>\mathcal{O}_L</math>.
The norm of an ideal <math>\mathfrak a</math> can be used to give an upper bound on the field norm of the smallest nonzero element it contains:
there always exists a nonzero <math>a\in\mathfrak a</math> for which
- <math>\left|N_{L/\mathbb{Q}}(a)\right|\leq \left ( \frac{2}{\pi}\right )^s \sqrt{\left|\Delta_L\right|}N(\mathfrak a),</math>
where
- <math>\Delta_L</math> is the discriminant of <math>L</math> and
- <math>s</math> is the number of pairs of (non-real) complex embeddings of Шаблон:Math into <math>\mathbb{C}</math> (the number of complex places of Шаблон:Math).[4]
See also
References
