Direct image functor

2013-11-11T15:29:31Z

145.97.30.152: /* Higher direct images */

{{Redirect|Brill's theorem|the result in algebraic geometry|Brill–Noether theorem}}

[[File:Discriminant49CubicFieldFundamentalDomain.png|thumb|300px|right|A fundamental domain of the ring of integers of the field ''K'' obtained from '''Q''' by adjoining a root of ''x''3 − ''x''2 − 2''x'' + 1. This fundamental domain sits inside ''K'' &otimes;'''Q''''''R'''. The discriminant of ''K'' is 49 = 72. Accordingly, the volume of the fundamental domain is 7 and ''K'' is only [[Splitting of prime ideals in Galois extensions|ramified]] at 7.]]
In [[mathematics]], the '''discriminant of an [[algebraic number field]]''' is a numerical [[invariant (mathematics)|invariant]] that, loosely speaking, measures the size of the ([[ring of integers]] of the) algebraic number field. More specifically, it is related to the volume of the [[fundamental domain]] of the ring of integers, and it regulates which [[prime number|primes]] are [[Ramified prime#In algebraic number theory|ramified]].

The discriminant is one of the most basic invariants of a number field, and occurs in several important [[Analytic Number Theory|analytic]] formulas such as the [[functional equation (L-function)|functional equation]] of the [[Dedekind zeta function]] of ''K'', and the [[analytic class number formula]] for ''K''. An old theorem of [[Charles Hermite|Hermite]] states that there are only finitely many number fields of bounded discriminant, however determining this quantity is still an [[open problem]], and the subject of current research.<ref>{{harvnb|Cohen|Diaz y Diaz|Olivier|2002}}</ref>

The discriminant of ''K'' can be referred to as the '''absolute discriminant''' of ''K'' to distinguish it from the '''relative discriminant''' of an [[field extension|extension]] ''K''/''L'' of number fields. The latter is an [[Ideal (ring theory)|ideal]] in the ring of integers of ''L'', and like the absolute discriminant it indicates which primes are ramified in ''K''/''L''. It is a generalization of the absolute discriminant allowing for ''L'' to be bigger than '''Q'''; in fact, when ''L'' = '''Q''', the relative discriminant of ''K''/'''Q''' is the [[principal ideal]] of '''Z''' generated by the absolute discriminant of ''K''.

==Definition==
Let ''K'' be an algebraic number field, and let ''O''''K'' be its ring of integers. Let ''b''1, ..., ''b''''n'' be an [[integral basis]] of ''O''''K'' (i.e. a basis as a [[Module (mathematics)|'''Z'''-module]]), and let {σ1, ..., σ''n''} be the set of embeddings of ''K'' into the [[complex number]]s (i.e. [[injective]] [[ring homomorphism]]s ''K'' → '''C'''). The '''discriminant''' of ''K'' is the [[Square (algebra)|square]] of the [[determinant]] of the ''n'' by ''n'' [[Matrix (mathematics)|matrix]] ''B'' whose (''i'',''j'')-entry is σ''i''(''b''''j''). Symbolically,
:<math>\Delta_K=\left(\operatorname{det}\left(\begin{array}{cccc}
\sigma_1(b_1) & \sigma_1(b_2) &\cdots & \sigma_1(b_n) \\
\sigma_2(b_1) & \ddots & & \vdots \\
\vdots & & \ddots & \vdots \\
\sigma_n(b_1) & \cdots & \cdots & \sigma_n(b_n)
\end{array}\right)\right)^2.
</math>
 
Equivalently, the [[Field trace|trace]] from ''K'' to '''Q''' can be used. Specifically, define the [[Trace form#Trace form and discriminant|trace form]] to be the matrix whose (''i'',''j'')-entry is
'''Tr'''''K''/'''Q'''(''b''''i''''b''''j''). This matrix equals ''B''T''B'', so the discriminant of ''K'' is the determinant of this matrix.

==Examples==
*[[Quadratic fields|Quadratic number fields]]: let ''d'' be a [[square-free integer]], then the discriminant of <math>K=\mathbf{Q}(\sqrt{d})</math> is<ref name=MP130/>
:: <math>\Delta_K=\left\{\begin{array}{ll} d &\text{if }d\equiv 1\pmod 4 \\ 4d &\text{if }d\equiv 2,3\pmod 4. \\\end{array}\right.</math>
:An integer that occurs as the discriminant of a quadratic number field is called a [[fundamental discriminant]].<ref>Definition 5.1.2 of {{harvnb|Cohen|1993}}</ref>
*[[Cyclotomic field]]s: let ''n'' > 2 be an integer, let ζ''n'' be a [[Root of unity|primitive ''n''th root of unity]], and let ''K''''n'' = '''Q'''(ζ''n'') be the ''n''th cyclotomic field. The discriminant of ''K''''n'' is given by<ref>Proposition 2.7 of {{harvnb|Washington|1997}}</ref><ref name=MP130>{{citation | first1=Yu. I. | last1=Manin | authorlink1=Yuri I. Manin | first2=A. A. | last2=Panchishkin | title=Introduction to Modern Number Theory | series=Encyclopaedia of Mathematical Sciences | volume=49 | edition=Second | year=2007 | isbn=978-3-540-20364-3 | issn=0938-0396 | zbl=1079.11002 | page=130 }}</ref>
:: <math>\Delta_{K_n} = (-1)^{\varphi(n)/2} \frac{n^{\varphi(n)}}{\displaystyle\prod_{p|n} p^{\varphi(n)/(p-1)}}</math>
: where <math>\varphi(n)</math> is [[Euler's totient function]], and the product in the denominator is over primes ''p'' dividing ''n''.
*Power bases: In the case where the ring of integers has a [[power integral basis]], that is, can be written as ''O''''K'' = '''Z'''[α], the discriminant of ''K'' is equal to the [[discriminant]] of the [[Minimal polynomial (field theory)|minimal polynomial]] of α. To see this, one can chose the integral basis of ''O''''K'' to be ''b''1 = 1, ''b''2 = α, ''b''3 = ''α''2, ..., ''b''''n'' = ''α''''n''−1. Then, the matrix in the definition is the [[Vandermonde matrix]] associated to α''i'' = σ''i''(α), whose determinant squared is
:: <math>\prod_{1\leq i<j\leq n}(\alpha_i-\alpha_j)^2</math>
:which is exactly the definition of the discriminant of the minimal polynomial.
*Let ''K'' = '''Q'''(α) be the number field obtained by [[Adjunction (field theory)|adjoining]] a [[Root of a function|root]] α of the [[polynomial]] ''x''3 − ''x''2 − 2''x'' − 8. This is [[Richard Dedekind]]'s original example of a number field whose ring of integers does not possess a power basis. An integral basis is given by {1, α, α(α + 1)/2} and the discriminant of ''K'' is −503.<ref>{{harvnb|Dedekind|1878}}, pp. 30–31</ref><ref>{{harvnb|Narkiewicz|2004}}, p. 64</ref>
*Repeated discriminants: the discriminant of a quadratic field uniquely identifies it, but this is not true, in general, for [[degree of a number field|higher-degree]] number fields. For example, there are two [[isomorphism|non-isomorphic]] [[cubic field]]s of discriminant 3969. They are obtained by adjoining a root of the polynomial {{nobreak|''x''3 − 21''x'' + 28}} or {{nobreak|''x''3 − 21''x'' − 35}}, respectively.<ref>{{harvnb|Cohen|1993|loc=Theorem 6.4.6}}</ref>


==Basic results==
*'''Brill's theorem''':<ref>{{harvnb|Koch|1997|p=11}}</ref> The [[sign (mathematics)|sign]] of the discriminant is (−1)''r''2 where ''r''2 is the number of [[Algebraic number field#Archimedean places|complex places]] of ''K''.<ref>Lemma 2.2 of {{harvnb|Washington|1997}}</ref>
*A prime ''p'' ramifies in ''K'' if, and only if, ''p'' divides Δ''K'' .<ref>Corollary III.2.12 of {{harvnb|Neukirch|1999}}</ref>
*'''Stickelberger's theorem''':<ref>Exercise I.2.7 of {{harvnb|Neukirch|1999}}</ref>
:: <math>\Delta_K\equiv 0\text{ or }1 \pmod 4.</math>
*'''[[Minkowski's bound]]''':<ref>Proposition III.2.14 of {{harvnb|Neukirch|1999}}</ref> Let ''n'' denote the [[Degree of a field extension|degree]] of the extension ''K''/'''Q''' and ''r''2 the number of complex places of ''K'', then
:: <math>|\Delta_K|^{1/2}\geq \frac{n^n}{n!}\left(\frac{\pi}{4}\right)^{r_2} \geq \frac{n^n}{n!}\left(\frac{\pi}{4}\right)^{n/2}.</math>

*'''Minkowski's theorem''':<ref>Theorem III.2.17 of {{harvnb|Neukirch|1999}}</ref> If ''K'' is not '''Q''', then |Δ''K''| > 1 (this follows directly from the Minkowski bound).
*'''[[Hermite–Minkowski theorem]]''':<ref>Theorem III.2.16 of {{harvnb|Neukirch|1999}}</ref> Let ''N'' be a positive integer. There are only finitely many (up to isomorphisms) algebraic number fields ''K'' with |Δ''K''| < ''N''. Again, this follows from the Minkowski bound together Hermite's theorem (There are only finitely many algebraic number fields with prescribed discriminant).

==History==
[[File:Dedekind.jpeg|thumb|150px|right|Richard Dedekind showed that every number field possesses an integral basis, allowing him to define the discriminant of an arbitrary number field.<ref name=DedekindX>Dedekind's supplement X of the second edition of [[Peter Gustav Lejeune Dirichlet]]'s [[Vorlesungen über Zahlentheorie]] {{harv|Dedekind|1871}}</ref>]]
The definition of the discriminant of a general algebraic number field, ''K'', was given by Dedekind in 1871.<ref name=DedekindX/> At this point, he already knew the relationship between the discriminant and ramification.<ref>{{harvnb|Bourbaki|1994}}</ref>

Hermite's theorem predates the general definition of the discriminant with Charles Hermite publishing a proof of it in 1857.{{sfn|Hermite|1857}} In 1877, [[Alexander von Brill]] determined the sign of the discriminant.{{sfn|Brill|1877}} [[Leopold Kronecker]] first stated Minkowski's theorem in 1882,{{sfn|Kronecker|1882}} though the first proof was given by Hermann Minkowski in 1891.{{sfn|Minkowski|1891a}} In the same year, Minkowski published his bound on the discriminant.{{sfn|Minkowski| 1891b}} Near the end of the nineteenth century, [[Ludwig Stickelberger]] obtained his theorem on the residue of the discriminant modulo four.{{sfn|Stickelberger|1897}}<ref>All facts in this paragraph can be found in {{harvnb|Narkiewicz|2004|pp=59, 81}}</ref>

==Relative discriminant==
The discriminant defined above is sometimes referred to as the ''absolute'' discriminant of ''K'' to distinguish it from the '''relative discriminant''' Δ''K''/''L'' of an extension of number fields ''K''/''L'', which is an ideal in ''O''''L''. The relative discriminant is defined in a fashion similar to the absolute discriminant, but must take into account that ideals in ''O''''L'' may not be principal and that there may not be an integral basis of ''K''/''L''. Let {σ1, ..., σ''n''} be the set of embeddings of ''K'' into '''C''' which are the identity on ''L''. If ''b''1, ..., ''b''''n'' is any basis of ''K'' over ''L'', let ''d''(''b''1, ..., ''b''''n'') be the square of the determinant of the ''n'' by ''n'' matrix whose (''i'',''j'')-entry is σ''i''(''b''''j''). Then, the relative discriminant of ''K''/''L'' is the ideal generated by the ''d''(''b''1, ..., ''b''''n'') as {''b''1, ..., ''b''''n''} varies over all bases of ''K''/''L'' with the property that ''bi'' ∈ ''OK'' for all ''i''. Alternatively, the relative discriminant of ''K''/''L'' is the [[ideal norm|norm]] of the [[different ideal|different]] of ''K''/''L''.<ref name="NeukirchIII2">{{harvnb|Neukirch|1999|loc=§III.2}}</ref> When ''L'' = '''Q''', the relative discriminant Δ''K''/'''Q''' is the principal ideal of '''Z''' generated by the absolute discriminant Δ''K'' . In a [[tower of fields]] ''K''/''L''/''F'' the relative discriminants are related by

:<math>\Delta_{K/F} = \mathcal{N}_{L/F}\left({\Delta_{K/L}}\right) \Delta_{L/F}^{[K:L]}</math>

where <math>\mathcal{N}</math> denotes relative [[field norm|norm]].<ref>Corollary III.2.10 of {{harvnb|Neukirch|1999}} or Proposition III.2.15 of {{harvnb|Fröhlich|Taylor|1993}}</ref>

===Ramification===
The relative discriminant regulates the [[Tame ramification|ramification]] data of the field extension ''K''/''L''. A prime ideal ''p'' of ''L'' ramifies in ''K'' if, and only if, it divides the relative discriminant Δ''K''/''L''. An extension is unramified if, and only if, the discriminant is the unit ideal.<ref name="NeukirchIII2"/> The Minkowski bound above shows that there are no non-trivial unramified extensions of '''Q'''. Fields larger than '''Q''' may have unramified extensions, for example, for any field with [[class number (number theory)|class number]] greater than one, its [[Hilbert class field]] is a non-trivial unramified extension.

==Root discriminant==
The '''root discriminant''' of a number field, ''K'', of degree ''n'', often denoted rd''K'', is defined as the ''n''-th root of the absolute value of the (absolute) discriminant of ''K''.<ref name=voight2008>{{harvnb|Voight|2008}}</ref> The relation between relative discriminants in a tower of fields shows that the root discriminant does not change in an unramified extension. The existence of a [[class field tower]] provides bounds on the root discriminant: the existence of an infinite class field tower over '''Q'''(√-m) where ''m'' = 3·5·7·11·19 shows that there are infinitely many fields with root discriminant 2√''m'' ≈ 296.276.<ref name=koch181>{{harvnb|Koch|1997|pp=181–182}}</ref> If we let ''r'' and 2''s'' be the number of real and complex embeddings, so that ''n'' = ''r'' + 2''s'', put ''ρ'' = ''r''/''n'' and ''σ'' = 2''s''/''n''. Set ''α''(''ρ'', ''σ'') to be the infimum of rd''K'' for ''K'' with (''r', 2''s') = (''ρn'', ''σn''). We have<ref name=koch181/>

:<math> \alpha(\rho,\sigma) \ge 60.8^\rho 22.3^\sigma </math>

and on the assumption of the [[generalized Riemann hypothesis]]

:<math> \alpha(\rho,\sigma) \ge 215.3^\rho 44.7^\sigma . </math>

So we have ''α''(0,1) < 296.276. Martinet has shown ''α''(0,1) < 93 and ''α''(1,0) < 1059.<ref name=koch181/><ref>{{cite journal | zbl=0369.12007 | last=Martinet | first=Jacques | title=Tours de corps de classes et estimations de discriminants | language=French | journal=[[Inventiones Mathematicae]] | volume=44 | pages=65–73 | year=1978 }}</ref> {{harvnb|Voight|2008}} proves that for totally real fields, the root discriminant is > 14, with 1229 exceptions.

==Relation to other quantities==
*When embedded into <math>K\otimes_\mathbf{Q}\mathbf{R}</math>, the volume of the fundamental domain of ''O''''K'' is <math>\sqrt{|\Delta_K|}</math> (sometimes a different [[Measure (mathematics)|measure]] is used and the volume obtained is <math>2^{-r_2}\sqrt{|\Delta_K|}</math>, where ''r''2 is the number of complex places of ''K'').
*Due to its appearance in this volume, the discriminant also appears in the functional equation of the Dedekind zeta function of ''K'', and hence in the analytic class number formula, and the [[Brauer–Siegel theorem]].
*The relative discriminant of ''K''/''L'' is the [[Artin conductor]] of the [[regular representation]] of the [[Galois group]] of ''K''/''L''. This provides a relation to the Artin conductors of the [[character group|characters]] of the Galois group of ''K''/''L'', called the [[conductor-discriminant formula]].<ref>Section 4.4 of {{harvnb|Serre|1967}}</ref>

==Notes==
{{reflist|2}}

==References==

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==Further reading==
* {{Citation | last=Milne | first=James S. | author-link=James S. Milne | title=Algebraic Number Theory | year=1998 | url=http://www.jmilne.org/math/CourseNotes/ant.html | accessdate=2008-08-20}}

{{DEFAULTSORT:Discriminant Of An Algebraic Number Field}}
[[Category:Algebraic number theory]]

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