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| In [[commutative algebra]], the '''norm of an ideal''' is a generalization of a [[field norm|norm]] of an element in the field extension. It is particularly important in number theory since it measures the size of an [[ideal (ring theory)|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''.
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| == Relative norm ==
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| Let ''A'' be a [[Dedekind domain]] with the [[field of fractions]] ''K'' and ''B'' be the [[integral closure]] of ''A'' in a finite [[separable extension]] ''L'' of ''K''. (In particular, ''B'' is Dedekind then.) Let <math>\operatorname{Id}(A)</math> and <math>\operatorname{Id}(B)</math> be the [[ideal group]]s of ''A'' and ''B'', respectively (i.e., the sets of [[fractional ideal]]s.) Following {{harv|Serre|1979}}, the '''norm map'''
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| :<math>N_{B/A}: \operatorname{Id}(B) \to \operatorname{Id}(A)</math>
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| is a homomorphism given by
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| :<math>N_{B/A}(\mathfrak q) = \mathfrak{p}^{[B/\mathfrak q : A/\mathfrak p]}, \mathfrak q \in \operatorname{Spec} B, \mathfrak q | \mathfrak p.</math>
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| If <math>L, K</math> are local fields, <math>N_{B/A}(\mathfrak{b})</math> is defined to be a fractional ideal generated by the set <math>\{ N_{L/K}(x) | x \in \mathfrak{b} \}.</math> This definition is equivalent to the above and is given in {{harv|Iwasawa|1986}}.
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| For <math>\mathfrak a \in \operatorname{Id}(A)</math>, one has <math>N_{B/A}(\mathfrak a B) = \mathfrak a^n</math> where <math>n = [L : K]</math>. The definition is thus also compatible with norm of an element: <math>N_{B/A}(xB) = N_{L/K}(x)A.</math><ref>Serre, 1. 5, Proposition 14.</ref>
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| Let <math>L/K</math> be a finite Galois extension of number fields with [[ring of integer|rings of integer]] <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 one has
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| :<math>N_{L/K}(I)=\mathcal{O}_K \cap\prod_{\sigma \in G}^{} \sigma (I)\, </math>
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| which is an ideal of <math>\mathcal{O}_K</math>. The norm of a [[principal ideal]] generated by ''α'' is the ideal generated by the [[field norm]] of ''α''.
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| The norm map is defined from the set of ideals of <math>\mathcal{O}_L</math> to the set of ideals of <math>\mathcal{O}_K</math>. It is reasonable to use integers as the range for <math>N_{L/\mathbf{Q}}\,</math> since '''Z''' has trivial [[ideal class group]]. This idea does not work in general since the class group may not be trivial.
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| == Absolute norm ==
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| Let <math>L</math> be a [[Algebraic number field|number field]] with ring of integers <math>\mathcal{O}_L</math>, and <math>\alpha</math> a nonzero ideal of <math>\mathcal{O}_L</math>. Then the norm of <math>\alpha</math> is defined to be
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| :<math>N(\alpha) =\left [ \mathcal{O}_L: \alpha\right ]=|\mathcal{O}_L/\alpha|.\,</math> | |
| By convention, the norm of the zero ideal is taken to be zero.
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| If <math>\alpha</math> is a principal ideal with <math>\alpha=(a)</math>, then <math>N(\alpha)=|N(a)|</math>. For proof, cf. Marcus, theorem 22c, pp65ff.
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| The norm is also [[Completely multiplicative function|completely multiplicative]] in that if <math>\alpha</math> and <math>\beta</math> are ideals of <math>\mathcal{O}_L</math>, then <math>N(\alpha\cdot\beta)=N(\alpha)N(\beta)</math>. For proof, cf. Marcus, theorem 22a, pp65ff.
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| The norm of an ideal <math>\alpha</math> can be used to bound the norm of some nonzero element <math>x\in \alpha</math> by the inequality
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| :<math>|N(x)|\leq \left ( \frac{2}{\pi}\right ) ^ {r_2} \sqrt{|\Delta_L|}N(\alpha)</math> | |
| where <math>\Delta_L</math> is the [[Discriminant of an algebraic number field|discriminant]] of <math>L</math> and <math>r_2</math> is the number of pairs of complex embeddings of <math>L</math> into <math>\mathbf{C}</math>.
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| ==See also==
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| *[[Dedekind zeta function]]
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| ==References==
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| {{reflist}}
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| *{{citation
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| |author=Iwasawa, Kenkichi
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| |title=Local class field theory
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| |series=Oxford Science Publications
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| |note=Oxford Mathematical Monographs
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| |publisher=The Clarendon Press Oxford University Press
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| |place=New York
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| |date=1986
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| |pages=viii+155
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| |isbn=0-19-504030-9
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| |mr=863740 (88b:11080)
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| }}
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| *{{citation
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| |author=Marcus, Daniel A.
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| |title=Number fields
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| |note=Universitext
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| |publisher=Springer-Verlag
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| |place=New York
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| |date=1977
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| |pages=viii+279
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| |isbn=0-387-90279-1
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| |mr=0457396 (56 #15601)
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| }}
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| *{{citation
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| |author=Serre, Jean-Pierre
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| |title=Local fields,
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| |series=Graduate Texts in Mathematics
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| |volume=67
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| |note=Translated from the French by Marvin Jay Greenberg
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| |publisher=Springer-Verlag
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| |place=New York
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| |date=1979
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| |pages=viii+241
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| |isbn=0-387-90424-7
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| |mr=554237 (82e:12016)
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| }}
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| [[Category:Algebraic number theory]]
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| [[Category:Commutative algebra]]
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| [[Category:Ideals]]
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