Ideal norm
title: "Ideal norm" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["algebraic-number-theory", "commutative-algebra", "ideals-(ring-theory)"] topic_path: "science/mathematics" source: "https://en.wikipedia.org/wiki/Ideal_norm" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0
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.{{citation |last=Janusz |first=Gerald J. |title=Algebraic number fields |edition=second |series=Graduate Studies in Mathematics |volume=7 |publisher=American Mathematical Society |place=Providence, Rhode Island |date=1996 |isbn=0-8218-0429-4 |mr=1362545 |at=Proposition I.8.2
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.{{citation |last=Serre |first=Jean-Pierre |title=Local Fields |authorlink1= Jean-Pierre Serre |series=Graduate Texts in Mathematics |volume=67 |translator-link=Marvin Greenberg |translator-first=Marvin Jay |translator-last1=Greenberg |publisher=Springer-Verlag |place=New York |date=1979 |isbn=0-387-90424-7 |mr=554237 |at=1.5, Proposition 14
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)= 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|.{{citation |last=Marcus |first=Daniel A. |title=Number fields |series=Universitext |publisher=Springer-Verlag |place=New York |date=1977 |isbn=0-387-90279-1 |mr=0457396 |at=Theorem 22c
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).
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).{{citation |first=Jürgen |last=Neukirch |title=Algebraic number theory |series=Grundlehren der mathematischen Wissenschaften |publisher=Springer-Verlag |place=Berlin |date=1999 |volume=322 |isbn=3-540-65399-6 |at=Lemma 6.2 |mr=1697859 |doi=10.1007/978-3-662-03983-0
References
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