en>K9re11 at 13:02, 8 July 2014

2014-07-08T13:02:45Z

← Older revision		Revision as of 15:02, 8 July 2014
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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''.~~		Eusebio Stanfill is what's blogged on my birth records although it is in no way the name on private birth certificate. Vermont could be where my home could be described as. Software rising has been my 24-hour period job for a while. To farrenheit is the only activity my wife doesn't approve of. You can unearth my website here: http://circuspartypanama.com<br><br>My site :: [http://circuspartypanama.com clash of clans bot]

	~~== Relative norm ==~~
	~~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'''~~
	~~:<math>N_{B/A}: \operatorname{Id}(B) \to \operatorname{Id}(A)</math>~~
	~~is a homomorphism given by~~
	~~:<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>~~

	~~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}}.~~

	~~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>~~

	~~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

	~~:<math>N_{L/K}(I)=\mathcal{O}_K \cap\prod_{\sigma \in G}^{} \sigma (I)\, </math>~~

	~~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''~~.

	~~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.~~

	~~== Absolute norm ==~~

	~~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~~
	:~~<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.~~

	~~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.~~

	~~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.~~

	~~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~~
	:~~<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>.~~

	~~==See also==~~
	*[[Dedekind zeta function]]

	~~==References==~~
	~~{{reflist}}~~
	*{{citation
	~~\|author=Iwasawa, Kenkichi~~
	~~\|title=Local class field theory~~
	~~\|series=Oxford Science Publications~~
	~~\|note=Oxford Mathematical Monographs~~
	~~\|publisher=The Clarendon Press Oxford University Press~~
	~~\|place=New York~~
	~~\|date=1986~~
	~~\|pages=viii+155~~
	~~\|isbn=0-19-504030-9~~
	~~\|mr=863740 (88b:11080)~~
	}}
	*{{citation
	~~\|author=Marcus, Daniel A.~~
	~~\|title=Number fields~~
	~~\|note=Universitext~~
	~~\|publisher=Springer-Verlag~~
	~~\|place=New York~~
	~~\|date=1977~~
	~~\|pages=viii+279~~
	~~\|isbn=0-387-90279-1~~
	~~\|mr=0457396 (56 #15601)~~
	}}
	*{{citation
	~~\|author=Serre, Jean-Pierre~~
	~~\|title=Local fields,~~
	~~\|series=Graduate Texts in Mathematics~~
	~~\|volume=67~~
	~~\|note=Translated from the French by Marvin Jay Greenberg~~
	~~\|publisher=Springer-Verlag~~
	~~\|place=New York~~
	~~\|date=1979~~
	~~\|pages=viii+241~~
	~~\|isbn=0-387-90424-7~~
	~~\|mr=554237 (82e:12016)~~
	}}

	~~[[Category:Algebraic number theory]]~~
	~~[[Category:Commutative algebra]]~~
	~~[[Category:Ideals]~~]

en>Jason Quinn: spacing fix

2013-07-01T03:12:16Z

spacing fix

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en>Eric Kvaalen: Made the first paragraph clearer.

2012-07-23T11:52:22Z

Made the first paragraph clearer.

New page

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← Older revision		Revision as of 15:02, 8 July 2014
Line 1:		Line 1:
	~~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''.~~		Eusebio Stanfill is what's blogged on my birth records although it is in no way the name on private birth certificate. Vermont could be where my home could be described as. Software rising has been my 24-hour period job for a while. To farrenheit is the only activity my wife doesn't approve of. You can unearth my website here: http://circuspartypanama.com<br><br>My site :: [http://circuspartypanama.com clash of clans bot]

	~~== Relative norm ==~~
	~~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'''~~
	~~:<math>N_{B/A}: \operatorname{Id}(B) \to \operatorname{Id}(A)</math>~~
	~~is a homomorphism given by~~
	~~:<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>~~

	~~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}}.~~

	~~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>~~

	~~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

	~~:<math>N_{L/K}(I)=\mathcal{O}_K \cap\prod_{\sigma \in G}^{} \sigma (I)\, </math>~~

	~~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''~~.

	~~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.~~

	~~== Absolute norm ==~~

	~~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~~
	:~~<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.~~

	~~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.~~

	~~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.~~

	~~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~~
	:~~<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>.~~

	~~==See also==~~
	*[[Dedekind zeta function]]

	~~==References==~~
	~~{{reflist}}~~
	*{{citation
	~~\|author=Iwasawa, Kenkichi~~
	~~\|title=Local class field theory~~
	~~\|series=Oxford Science Publications~~
	~~\|note=Oxford Mathematical Monographs~~
	~~\|publisher=The Clarendon Press Oxford University Press~~
	~~\|place=New York~~
	~~\|date=1986~~
	~~\|pages=viii+155~~
	~~\|isbn=0-19-504030-9~~
	~~\|mr=863740 (88b:11080)~~
	}}
	*{{citation
	~~\|author=Marcus, Daniel A.~~
	~~\|title=Number fields~~
	~~\|note=Universitext~~
	~~\|publisher=Springer-Verlag~~
	~~\|place=New York~~
	~~\|date=1977~~
	~~\|pages=viii+279~~
	~~\|isbn=0-387-90279-1~~
	~~\|mr=0457396 (56 #15601)~~
	}}
	*{{citation
	~~\|author=Serre, Jean-Pierre~~
	~~\|title=Local fields,~~
	~~\|series=Graduate Texts in Mathematics~~
	~~\|volume=67~~
	~~\|note=Translated from the French by Marvin Jay Greenberg~~
	~~\|publisher=Springer-Verlag~~
	~~\|place=New York~~
	~~\|date=1979~~
	~~\|pages=viii+241~~
	~~\|isbn=0-387-90424-7~~
	~~\|mr=554237 (82e:12016)~~
	}}

	~~[[Category:Algebraic number theory]]~~
	~~[[Category:Commutative algebra]]~~
	~~[[Category:Ideals]~~]

Ring of integers - Revision history

en>K9re11 at 13:02, 8 July 2014

en>Jason Quinn: spacing fix

en>Eric Kvaalen: Made the first paragraph clearer.