|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| In [[number theory]], the '''class number formula''' relates many important invariants of a [[number field]] to a special value of its [[Dedekind zeta function]]
| | The title of the author is Jayson. He is an info officer. My spouse and I [http://Www.Article-galaxy.com/profile.php?a=143251 live psychic reading] in Kentucky. What me and my family adore is doing ballet but I've been using on new issues lately. |
| | |
| ==General statement of the class number formula==
| |
| Let ''K'' be a number field with [''K'':'''Q''']=''n''=''r''<sub>1</sub>+2''r''<sub>2</sub>, where <math>r_1</math> denotes the number of [[real and complex embeddings|real embeddings]] of ''K'', and <math>2r_2</math> is the number of complex embeddings of ''K''. Let
| |
| | |
| :<math> \zeta_K(s) \,</math>
| |
| | |
| be the Dedekind zeta function of ''K''. Also define the following [[Invariant (mathematics)|invariants]]:
| |
| *<math>h_K</math> is the [[ideal class|class number]], the number of elements in the ideal class group of ''K''.
| |
| *<math>\operatorname{Reg}_K</math> is the [[regulator (mathematics)|regulator]] of ''K''.
| |
| *<math>w_K</math> is the number of [[root of unity|roots of unity]] contained in ''K''.
| |
| *<math>D_K</math> is the [[discriminant of an algebraic number field|discriminant]] of the [[Algebraic extension|extension]] ''K''/'''Q'''.
| |
| Then:
| |
| | |
| '''Theorem 1''' (Class Number Formula) ''The [[Dedekind zeta function]] of ''K'', <math>\zeta_K(s)</math> [[conditionally convergent|converges absolutely]] for <math>\Re(s)>1</math> and extends to a [[meromorphic]] [[function (mathematics)|function]] defined for all complex ''s'' with only one [[simple pole]] at ''s'' = 1, with residue''
| |
| | |
| :<math> \lim_{s\to 1} (s-1)\zeta_K(s)=\frac{2^{r_1}\cdot(2\pi)^{r_2}\cdot h_K\cdot \operatorname{Reg}_K}{w_K \cdot \sqrt{|D_K|}}</math>
| |
| | |
| This is the most general "class number formula". In particular cases, for example when ''K'' is a [[cyclotomic extension]] of '''Q''', there are particular and more refined class number formulas.
| |
| | |
| ==Proof==
| |
| | |
| The idea of the proof of the class number formula is most easily seen when ''K'' = '''Q'''(i). In this case, the ring of integers in ''K'' is the [[Gaussian integer]]s.
| |
| | |
| An elementary manipulation shows that the residue of the Dedekind zeta function at ''s'' = 1 is the average of the coefficients of the [[Dirichlet series]] representation of the Dedekind zeta function. The ''n''<sup>th</sup> coefficient of the Dirichlet series is essentially the number of representations of ''n'' as a sum of two squares of nonnegative integers. So one can compute the residue of the Dedekind zeta function at ''s'' = 1 by computing the average number of representations. As in the article on the [[Gauss circle problem]], one can compute this by approximating the number of lattice points inside of a quarter circle centered at the origin, concluding that the residue is one quarter of pi.
| |
| | |
| The proof when ''K'' is an arbitrary imaginary quadratic number field is very similar.<ref>https://www.math.umass.edu/~weston/oldpapers/cnf.pdf</ref>
| |
| | |
| In the general case, by [[Dirichlet's unit theorem]], the group of units in the ring of integers of ''K'' is infinite. One can nevertheless reduce the computation of the residue to a lattice point counting problem using the classical theory of real and complex embeddings<ref>http://planetmath.org/realandcomplexembeddings</ref> and approximate the number of lattice points in a region by the volume of the region, to complete the proof.
| |
| | |
| ==Dirichlet class number formula==
| |
| | |
| [[Peter Gustav Lejeune Dirichlet]] published a proof of the class number formula for [[quadratic field]]s in 1839, but it was stated in the language of [[quadratic forms]] rather than classes of [[Ideal (ring theory)|ideal]]s. It appears that Gauss already knew this formula in 1801.<ref>http://mathoverflow.net/questions/109330/did-gauss-know-dirichlets-class-number-formula-in-1801</ref>
| |
| | |
| This exposition follows Davenport.<ref name=Davenport>
| |
| {{cite book |last1=Davenport |first1=Harold |authorlink1=Harold Davenport |editor1-first=Hugh L. |editor1-last=Montgomery |editor1-link=Hugh Montgomery (mathematician) |title=Multiplicative Number Theory |url=http://books.google.com/books?id=U91lsCaJJmsC |accessdate=2009-05-26 |edition=3rd |series=Graduate Texts in Mathematics |volume=74 |year=2000|publisher=Springer-Verlag |location=New York |isbn=978-0-387-95097-6 |pages=43–53 }}
| |
| </ref>
| |
| | |
| Let ''d'' be a [[fundamental discriminant]], and write ''h(d)'' for the number of equivalence classes of quadratic forms with discriminant ''d''. Let <math>\chi = \left(\!\frac{d}{m}\!\right)</math> be the [[Kronecker symbol]]. Then <math>\chi</math> is a [[Dirichlet character]]. Write <math>L(s,\chi)</math> for the [[Dirichlet L-series]] based on <math>\chi</math>. For ''d > 0'', let ''t > 0'', ''u > 0'' be the solution to the [[Pell equation]] <math>t^2 - d u^2 = 4</math> for which ''u'' is smallest, and write
| |
| :<math>\epsilon = \frac{1}{2}(t + u \sqrt{d}).</math>
| |
| (Then ε is either a [[fundamental unit (number theory)|fundamental unit]] of the [[real quadratic field]] <math>\mathbb{Q}(\sqrt{d})</math> or the square of a fundamental unit.)
| |
| For ''d'' < 0, write ''w'' for the number of automorphs of quadratic forms of discriminant ''d''; that is,
| |
| :<math>w =
| |
| \begin{cases}
| |
| 2, & d < -4; \\
| |
| 4, & d = -4; \\
| |
| 6, & d = -3.
| |
| \end{cases}
| |
| </math>
| |
| Then Dirichlet showed that
| |
| :<math>h(d)=
| |
| \begin{cases}
| |
| \dfrac{w \sqrt{|d|}}{2 \pi} L(1,\chi), & d < 0; \\
| |
| \dfrac{\sqrt{d}}{\ln \epsilon} L(1,\chi), & d > 0.
| |
| \end{cases}</math>
| |
| This is a special case of Theorem 1 above: for a [[quadratic field]] ''K'', the Dedekind zeta function is just <math>\zeta_K(s) = \zeta(s) L(s, \chi)</math>, and the residue is <math>L(1,\chi)</math>. Dirichlet also showed that the ''L''-series can be written in a finite form, which gives a finite form for the class number. Suppose <math>\chi</math> is [[Dirichlet character#Primitive characters and conductor|primitive]] with prime [[Dirichlet character#Primitive characters and conductor|conductor]] <math>q</math>. Then
| |
| :<math> L(1, \chi) =
| |
| \begin{cases}
| |
| -\dfrac{\pi}{q^{3/2}}\sum_{m=1}^{q-1} m \left( \dfrac{m}{q} \right), & q \equiv 3 \mod 4; \\
| |
| -\dfrac{1}{q^{1/2}}\sum_{m=1}^{q-1} \left( \dfrac{m}{q} \right) \ln 2\sin \dfrac{m\pi}{q} , & q \equiv 1 \mod 4.
| |
| \end{cases}</math>
| |
| | |
| ==Galois extensions of the rationals==
| |
| If ''K'' is a [[Galois extension]] of '''Q''', the theory of [[Artin L-function]]s applies to <math> \zeta_K(s)</math>. It has one factor of the [[Riemann zeta function]], which has a pole of residue one, and the quotient is regular at ''s'' = 1. This means that the right-hand side of the class number formula can be equated to a left-hand side
| |
| | |
| :Π ''L''(1,ρ)<sup>dim ρ</sup>
| |
| | |
| with ρ running over the classes of irreducible non-trivial complex [[linear representation]]s of Gal(''K''/'''Q''') of dimension dim(ρ). That is according to the standard decomposition of the [[regular representation]].
| |
| | |
| ==Abelian extensions of the rationals==
| |
| This is the case of the above, with Gal(''K''/'''Q''') an [[abelian group]], in which all the ρ can be replaced by [[Dirichlet character]]s (via [[class field theory]]) for some modulus ''f'' called the [[conductor of an abelian extension|conductor]]. Therefore all the ''L''(1) values occur for [[Dirichlet L-function]]s, for which there is a classical formula, involving logarithms.
| |
| | |
| By the [[Kronecker–Weber theorem]], all the values required for an '''analytic class number formula''' occur already when the cyclotomic fields are considered. In that case there is a further formulation possible, as shown by [[Ernst Kummer|Kummer]]. The regulator, a calculation of volume in 'logarithmic space' as divided by the logarithms of the units of the cyclotomic field, can be set against the quantities from the ''L''(1) recognisable as logarithms of [[cyclotomic unit]]s. There result formulae stating that the class number is determined by the index of the cyclotomic units in the whole group of units.
| |
| | |
| In [[Iwasawa theory]], these ideas are further combined with [[Stickelberger's theorem]].
| |
| | |
| ==Notes==
| |
| {{reflist}}
| |
| | |
| ==References==
| |
| * {{cite book | author=W. Narkiewicz | title=Elementary and analytic theory of algebraic numbers | edition=2nd ed | publisher=[[Springer-Verlag]]/[[Polish Scientific Publishers PWN]] | year=1990 | isbn=3-540-51250-0 | pages=324–355 }}
| |
| | |
| {{PlanetMath attribution|id=4664|title=Class number formula}}
| |
| | |
| {{L-functions-footer}}
| |
| | |
| [[Category:Algebraic number theory]]
| |
| [[Category:Quadratic forms]]
| |
The title of the author is Jayson. He is an info officer. My spouse and I live psychic reading in Kentucky. What me and my family adore is doing ballet but I've been using on new issues lately.