Ideal norm

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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[edit]

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.

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[edit]

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 ]=|\mathcal{O}_L/\mathfrak a|.\,

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)=|N_{L/\mathbb{Q}}(a)|.[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).[4] 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

|N_{L/\mathbb{Q}}(a)|\leq \left ( \frac{2}{\pi}\right )^s \sqrt{|\Delta_L|}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).[5]

See also[edit]

References[edit]

  1. ^ Janusz, Proposition I.8.2
  2. ^ Serre, 1.5, Proposition 14.
  3. ^ Marcus, Theorem 22c, pp. 65-66.
  4. ^ Marcus, Theorem 22a, pp. 65-66
  5. ^ Neukirch, Lemma 6.2