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In [[algebraic number theory]], '''Minkowski's bound''' gives an [[upper bound]] of the norm of ideals to be checked in order to determine the [[Ideal class group|class number]] of a [[number field]] ''K''. It is named for the mathematician [[Hermann Minkowski]]. | |||
Let ''D'' be the [[Discriminant of an algebraic number field|discriminant]] of the field, ''n'' be the degree of ''K'' over <math>\mathbb{Q}</math>, and <math>2 r_2 = n - r_1</math> be the number of [[complex embedding]]s where <math>r_1</math> is the number of [[real embedding]]s. Then every class in the [[ideal class group]] of ''K'' contains an [[integral ideal]] of [[Ideal norm|norm]] not exceeding Minkowski's bound | |||
:<math> M_K = \sqrt{|D|} \left(\frac{4}{\pi}\right)^{r_2} \frac{n!}{n^n} \ . </math> | |||
'''Minkowski's constant''' for the field ''K'' is this bound ''M''<sub>''K''</sub>.<ref name=PZ384>Pohst & Zassenhaus (1989) p.384</ref> | |||
Since the number of integral ideals of given norm is finite, the finiteness of the class number is an immediate consequence,<ref name=PZ384/> and further, the [[ideal class group]] is generated by the [[prime ideal]]s of norm at most ''M''<sub>''K''</sub>. | |||
The result is a consequence of [[Minkowski's theorem]]. | |||
Minkowski's bound may be used to derive a lower bound for the discriminant of a field ''K'' given ''n'', ''r''<sub>1</sub> and ''r''<sub>2</sub>. Since an integral ideal has norm at least one, we have 1 ≤ ''M''<sub>''K''</sub>, so that | |||
:<math> \sqrt{|D|} \ge \left(\frac{\pi}{4}\right)^{r_2} \frac{n^n}{n!} \ge \left(\frac{\pi}{4}\right)^{n/2} \frac{n^n}{n!} \ . </math> | |||
For ''n'' at least 2, it is easy to show that the lower bound is greater than 1, so we obtain '''Minkowski's Theorem''', that the discriminant of every number field, other than '''Q''', is non-trivial. This implies that the field of rational numbers has no [[unramified extension]]. | |||
== References == | |||
{{reflist}} | |||
* {{cite book | first=Helmut | last=Koch | title=Algebraic Number Theory | publisher=[[Springer-Verlag]] | year=1997 | isbn=3-540-63003-1 | zbl=0819.11044 | series=Encycl. Math. Sci. | volume=62 | edition=2nd printing of 1st }} | |||
* {{cite book |last=Lang |first=Serge |title=Algebraic Number Theory |authorlink=Serge Lang |edition=second |year=1994 | series=[[Graduate Texts in Mathematics]] | volume=110 | publisher=Springer |location=New York |isbn=0-387-94225-4 | zbl=0811.11001 }} | |||
* {{cite book | last1=Pohst | first1=M. | last2=Zassenhaus | first2=H. | author2-link=Hans Zassenhaus | title=Algorithmic Algebraic Number Theory | series=Encyclopedia of Mathematics and its Applications | volume=30 | publisher=[[Cambridge University Press]] | year=1989 | isbn=0-521-33060-2 | zbl=0685.12001 }} | |||
==External links== | |||
* {{Planetmath reference|id=6822|title=Using Minkowski's Constant To Find A Class Number}} | |||
*Stevenhagen, Peter. [http://websites.math.leidenuniv.nl/algebra/ant.pdf ''Number Rings''.] | |||
*[http://sbseminar.wordpress.com/2007/08/16/the-minkowski-bound/ The Minkowski Bound] at Secret Blogging Seminar | |||
{{Numtheory-stub}} | |||
[[Category:Algebraic number theory]] | |||
Revision as of 19:24, 20 April 2013
In algebraic number theory, Minkowski's bound gives an upper bound of the norm of ideals to be checked in order to determine the class number of a number field K. It is named for the mathematician Hermann Minkowski.
Let D be the discriminant of the field, n be the degree of K over , and be the number of complex embeddings where is the number of real embeddings. Then every class in the ideal class group of K contains an integral ideal of norm not exceeding Minkowski's bound
Minkowski's constant for the field K is this bound MK.[1]
Since the number of integral ideals of given norm is finite, the finiteness of the class number is an immediate consequence,[1] and further, the ideal class group is generated by the prime ideals of norm at most MK.
The result is a consequence of Minkowski's theorem.
Minkowski's bound may be used to derive a lower bound for the discriminant of a field K given n, r1 and r2. Since an integral ideal has norm at least one, we have 1 ≤ MK, so that
For n at least 2, it is easy to show that the lower bound is greater than 1, so we obtain Minkowski's Theorem, that the discriminant of every number field, other than Q, is non-trivial. This implies that the field of rational numbers has no unramified extension.
References
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- 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
External links
- Template:Planetmath reference
- Stevenhagen, Peter. Number Rings.
- The Minkowski Bound at Secret Blogging Seminar