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| In [[mathematics]], specifically the area of [[algebraic number theory]], a '''cubic field''' is an [[algebraic number field]] of [[Degree of a number field|degree]] three.
| | by Nas, is very fitting and the film agrees with it. Medical word press themes give you the latest medical designs. Wordpress Content management systems, being customer friendly, can be used extensively to write and manage websites and blogs. Word - Press also provides protection against spamming, as security is a measure issue. Also our developers are well convergent with the latest technologies and bitty-gritty of wordpress website design and promises to deliver you the best solution that you can ever have. <br><br>Creating a website from scratch can be such a pain. If a newbie missed a certain part of the video then they could always rewind. A Wordpress plugin is a software that you can install into your Wordpress site. You can up your site's rank with the search engines by simply taking a bit of time with your site. Now a days it has since evolved into a fully capable CMS platform which make it, the best platform in the world for performing online business. <br><br>But before choosing any one of these, let's compare between the two. s cutthroat competition prevailing in the online space won. We can active Akismet from wp-admin > Plugins > Installed Plugins. Every single Theme might be unique, providing several alternatives for webpage owners to reap the benefits of in an effort to instantaneously adjust their web page appear. If you have any questions on starting a Word - Press food blog or any blog for that matter, please post them below and I will try to answer them. <br><br>Additionally Word - Press add a default theme named Twenty Fourteen. As an example, if you are promoting a product that cures hair-loss, you most likely would not wish to target your adverts to teens. The templates are designed to be stand alone pages that have a different look and feel from the rest of your website. Can you imagine where you would be now if someone in your family bought an original painting from van Gogh during his lifetime. Here is more information regarding [http://ammi.me/backup_plugin_944922 wordpress dropbox backup] have a look at our own web site. Digital digital cameras now function gray-scale configurations which allow expert photographers to catch images only in black and white. <br><br>Someone with a basic knowledge of setting up a website should be able to complete the process in a couple of minutes however even basic users should find they are able to complete the installation in around 20 minutes by following the step by step guide online. Here's a list of some exciting Word - Press features that have created waves in the web development industry:. Must being, it's beneficial because I don't know about you, but loading an old website on a mobile, having to scroll down, up, and sideways' I find links being clicked and bounced around like I'm on a freaking trampoline. Web developers and newbies alike will have the ability to extend your web site and fit other incredible functions with out having to spend more. Press CTRL and the numbers one to six to choose your option. |
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| ==Definition==
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| If ''K'' is a [[field extension]] of the rational numbers '''Q''' of [[Degree of a field extension|degree]] [''K'':'''Q'''] = 3, then ''K'' is called a '''cubic field'''. Any such field is isomorphic to a field of the form
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| :<math>\mathbf{Q}[x]/(f(x))</math>
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| where ''f'' is an [[irreducible]] [[Degree of a polynomial|cubic]] [[polynomial]] with coefficients in '''Q'''. If ''f'' has three [[real number|real]] [[root of a polynomial|roots]], then ''K'' is called a '''totally real cubic field''' and it is an example of a [[totally real field]]. If, on the other hand, ''f'' has a non-real root, then ''K'' is called a '''complex cubic field'''.
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| A cubic field ''K'' is called a '''cyclic cubic field''', if it contains all three roots of its generating polynomial ''f''. Equivalently, ''K'' is a cyclic cubic field if it is a [[Galois extension]] of '''Q''', in which case its [[Galois group]] over '''Q''' is [[cyclic group|cyclic]] of [[order of a group|order]] three. This can only happen if ''K'' is totally real. It is a rare occurrence in the sense that if the set of cubic fields is ordered by [[Discriminant of an algebraic number field|discriminant]], then the proportion of cubic fields which are cyclic approaches zero as the bound on the discriminant approaches infinity.<ref>Harvey Cohn computed an asymptotic for the number of cyclic cubic fields {{harv|Cohn|1954}}, while [[Harold Davenport]] and [[Hans Heilbronn]] computed the asymptotic for all cubic fields {{harv|Davenport|Heilbronn|1971}}.</ref>
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| A cubic field is called a '''pure cubic field''', if it can be obtained by adjoining the real cube root <math>\sqrt[3]{n}</math> of a cubefree positive integer ''n'' to the rational number field '''Q'''.
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| ==Examples==
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| *Adjoining the real cube root of 2 to the rational numbers gives the cubic field <math>\mathbf{Q}(\sqrt[3]{2})</math>. This is an example of a pure cubic field, and hence of a complex cubic field. In fact, of all pure cubic fields, it has the smallest discriminant (in [[absolute value]]), namely −108.<ref>{{harvnb|Cohen|1993|loc=§B.3}} contains a table of complex cubic fields</ref>
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| *The complex cubic field obtained by adjoining to '''Q''' a root of {{nowrap|''x''<sup>3</sup> + ''x''<sup>2</sup> − 1}} is not pure. It has the smallest discriminant (in absolute value) of all cubic fields, namely −23.<ref>{{harvnb|Cohen|1993|loc=§B.3}}</ref>
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| *Adjoining a root of {{nowrap|''x''<sup>3</sup> + ''x''<sup>2</sup> − 2''x'' − 1}} to '''Q''' yields a cyclic cubic field, and hence a totally real cubic field. It has the smallest discriminant of all totally real cubic fields, namely 49.<ref>{{harvnb|Cohen|1993|loc=§B.4}} contains a table of totally real cubic fields and indicates which are cyclic</ref>
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| *The field obtained by adjoining to '''Q''' a root of {{nowrap|''x''<sup>3</sup> + ''x''<sup>2</sup> − 3''x'' − 1}} is an example of a totally real cubic field that is not cyclic. Its discriminant is 148, the smallest discriminant of a non-cyclic totally real cubic field.<ref>{{harvnb|Cohen|1993|loc=§B.4}}</ref>
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| *No [[cyclotomic field]]s are cubic because the degree of a cyclotomic field is equal to φ(''n''), where φ is [[Euler's totient function]], which only takes on even values (except for φ(1) = φ(2) = 1).
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| ==Galois closure==
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| A cyclic cubic field ''K'' is its own [[Galois closure]] with Galois group Gal(''K''/'''Q''') isomorphic to the cyclic group of order three. However, any other cubic field ''K'' is a non-galois extension of '''Q''' and has a field extension ''N'' of degree two as its Galois closure. The Galois group Gal(''N''/'''Q''') is isomorphic to the [[symmetric group]] ''S''<sub>3</sub> on three letters.
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| ==Associated quadratic field==
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| The discriminant of a cubic field ''K'' can be written uniquely as ''df''<sup>2</sup> where ''d'' is a [[fundamental discriminant]]. Then, ''K'' is cyclic if, and only if, ''d'' = 1, in which case the only subfield of ''K'' is '''Q''' itself. If ''d'' ≠ 1, then the Galois closure ''N'' of ''K'' contains a unique [[quadratic field]] ''k'' whose discriminant is ''d'' (in the case ''d'' = 1, the subfield '''Q''' is sometimes considered as the "degenerate" quadratic field of discriminant 1). The [[conductor (class field theory)|conductor]] of ''N'' over ''k'' is ''f'', and ''f''<sup>2</sup> is the [[relative discriminant]] of ''N'' over ''k''. The discriminant of ''N'' is ''d''<sup>3</sup>''f''<sup>4</sup>.<ref>{{harvnb|Hasse|1930}}</ref><ref name="Cohen 1993 loc=§6.4.5">{{harvnb|Cohen|1993|loc=§6.4.5}}</ref>
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| The field ''K'' is a pure cubic field if, and only if, ''d'' = −3. This is the case for which the quadratic field contained in the Galois closure of ''K'' is the cyclotomic field of cube roots of unity.<ref name="Cohen 1993 loc=§6.4.5"/>
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| ==Discriminant==
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| [[Image:PlotDiscriminantsOfRealCubicFields.svg|250px|right|thumb|The blue crosses are the number of totally real cubic fields of bounded discriminant. The black line is the asymptotic distribution to first order whereas the green line includes the second order term.<ref name="discdata">The exact counts were computed by Michel Olivier and are available at [http://pari.math.u-bordeaux.fr/pub/pari/packages/nftables/]. The first-order asymptotic is due to [[Harold Davenport]] and [[Hans Heilbronn]] {{harv|Davenport|Heilbronn|1971}}. The second-order term was conjectured by David P. Roberts {{harv|Roberts|2001}} and a proof has been announced by [[Manjul Bhargava]], Arul Shankar, and Jacob Tsimerman {{harv|Bhargava|Shankar|Tsimerman|2010}}.</ref>]]
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| [[Image:PlotDiscriminantsOfComplexCubicFields.svg|250px|right|thumb|The blue crosses are the number of complex cubic fields of bounded discriminant. The black line is the asymptotic distribution to first order whereas the green line includes the second order term.<ref name="discdata"/>]]
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| Since the sign of the [[Discriminant of an algebraic number field|discriminant]] of a number field ''K'' is (−1)<sup>''r''<sub>2</sub></sup>, where ''r''<sub>2</sub> is the number of conjugate pairs of complex embeddings of ''K'' into '''C''', the discriminant of a cubic field will be positive precisely when the field is totally real, and negative if it is a complex cubic field.
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| Given some real number ''N'' > 0 there are only finitely many cubic fields ''K'' whose discriminant ''D''<sub>''K''</sub> satisfies |''D''<sub>''K''</sub>| ≤ ''N''.<ref>[[Hermann Minkowski|H. Minkowski]], ''Diophantische Approximationen'', chapter 4, §5.</ref> Formulae are known which calculate the prime decomposition of ''D''<sub>''K''</sub>, and so it can be explicitly calculated.<ref>{{Cite journal |first=P. |last=Llorente |first2=E. |last2=Nart |title=Effective determination of the decomposition of the rational primes in a cubic field |journal=Proceedings of the American Mathematical Society |volume=87 |issue=4 |year=1983 |pages=579–585 |doi=10.1090/S0002-9939-1983-0687621-6 }}</ref>
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| However, it should be pointed out that, different from quadratic fields, several non-isomorphic cubic fields ''K''<sub>1</sub>, ..., ''K<sub>m</sub>'' may share the same discriminant ''D''. The number ''m'' of these fields is called the '''multiplicity'''<ref>{{Cite journal |first=D. C. |last=Mayer |title=Multiplicities of dihedral discriminants |journal=[[Mathematics of Computation|Math. Comp.]] |volume=58 |issue=198 |year=1992 |pages=831–847 and S55–S58 |doi=10.1090/S0025-5718-1992-1122071-3 }}</ref> of the discriminant ''D''. Some small examples are ''m'' = 2 for ''D'' = −1836,3969, ''m'' = 3 for ''D'' = −1228,22356, ''m'' = 4 for ''D'' = −3299,32009, and ''m'' = 6 for ''D'' = −70956,3054132.
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| Any cubic field ''K'' will be of the form ''K'' = '''Q'''(θ) for some number θ that is a root of the irreducible polynomial
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| :<math>f(X)=X^3-aX+b</math>
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| with ''a'' and ''b'' both being integers. The [[discriminant]] of ''f'' is Δ = 4''a''<sup>3</sup> − 27''b''<sup>2</sup>. Denoting the discriminant of ''K'' by ''D'', the '''index''' ''i''(θ) of θ is then defined by Δ = ''i''(θ)<sup>2</sup>''D''. | |
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| In the case of a non-cyclic cubic field ''K'' this index formula can be combined with the conductor formula ''D'' = ''f''<sup>2</sup>''d'' to obtain a decomposition of the polynomial discriminant Δ = ''i''(θ)<sup>2</sup>''f''<sup>2</sup>''d'' into the square of the product ''i''(θ)''f'' and the discriminant ''d'' of the quadratic field ''k'' associated with the cubic field ''K'', where ''d'' is squarefree up to a possible factor 2<sup>2</sup> or 2<sup>3</sup>. [[Georgy Voronoy]] gave a method for separating ''i''(θ) and ''f'' in the square part of Δ.<ref>G. F. Voronoi, ''Concerning algebraic integers derivable from a root of an equation of the third degree'', Master's Thesis, St. Petersburg, 1894 (Russian).</ref>
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| The study of the number of cubic fields whose discriminant is less than a given bound is a current area of research. Let ''N''<sup>+</sup>(''X'') (respectively ''N''<sup>−</sup>(''X'')) denote the number of totally real (respectively complex) cubic fields whose discriminant is bounded by ''X'' in absolute value. In the early 1970s, [[Harold Davenport]] and [[Hans Heilbronn]] determined the first term of the asymptotic behaviour of ''N''<sup>±</sup>(''X'') (i.e. as ''X'' goes to infinity).<ref>{{harvnb|Davenport|Heilbronn|1971}}</ref><ref>Their work can also be interpreted as a computation of the average size of the [[torsion (algebra)|3-torsion]] part of the [[class group]] of a [[quadratic field]], and thus constitutes one of the few proven cases of the [[Cohen–Lenstra conjectures]].{{citation needed|date=June 2010}}</ref> By means of an analysis of the [[residue (complex analysis)|residue]] of the [[Shintani zeta function]], combined with a study of the tables of cubic fields compiled by Karim Belabas {{harv|Belabas|1997}} and some [[heuristic]]s, David P. Roberts conjectured a more precise asymptotic formula:<ref>{{harvnb|Roberts|2001|loc=Conjecture 3.1}}</ref>
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| :<math>N^\pm(X)\sim\frac{A_\pm}{12\zeta(3)}X+\frac{4\zeta(\frac{1}{3})B_\pm}{5\Gamma(\frac{2}{3})^3\zeta(\frac{5}{3})}X^{\frac{5}{6}}</math>
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| where ''A''<sub>±</sub> = 1 or 3, ''B''<sub>±</sub> = 1 or <math>\sqrt{3}</math>, according to the totally real or complex case, ζ(''s'') is the [[Riemann zeta function]], and Γ(''s'') is the [[Gamma function]]. A proof of this formula has been announced by {{harvtxt|Bhargava|Shankar|Tsimerman|2010}} using methods based on Bhargava's earlier work, as well as {{harvtxt|Taniguchi|Thorne|2011}} based on the Shintani zeta function.
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| ==Unit group==
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| According to [[Peter Gustav Lejeune Dirichlet]], the torsionfree unit rank ''r'' of an algebraic number field ''K'' with ''r''<sub>1</sub> real embeddings and ''r''<sub>2</sub> pairs of conjugate complex embeddings is determined by the formula ''r'' = ''r''<sub>1</sub> + ''r''<sub>2</sub> − 1. Hence a totally real cubic field ''K'' with ''r''<sub>1</sub> = 3, ''r''<sub>2</sub> = 0 has two independent units ε<sub>1</sub>, ε<sub>2</sub> and a complex cubic field ''K'' with ''r''<sub>1</sub> = ''r''<sub>2</sub> = 1 has a single fundamental unit ε<sub>1</sub>. These fundamental systems of units can be calculated by means of generalized continued fraction algorithms by [[Georgy Voronoy|Voronoi]],<ref>{{cite book |first=G. F. |last=Voronoi |title=On a generalization of the algorithm of continued fractions |publisher=Doctoral Dissertation |location=Warsaw |year=1896 |language=Russian }}</ref> which have been interpreted geometrically by [[Boris Delaunay|Delone]] and [[Dmitry Faddeev|Faddeev]].<ref>{{cite book |first=B. N. |last=Delone |first2=D. K. |last2=Faddeev |title=The theory of irrationalities of the third degree |series=Translations of Mathematical Monographs |volume=10 |publisher=American Mathematical Society |location=Providence, Rhode Island |year=1964 }}</ref>
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| ==Notes==
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| {{Reflist|30em}}
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| ==References==
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| *Şaban Alaca, Kenneth S. Williams, ''Introductory algebraic number theory'', [[Cambridge University Press]], 2004.
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| *{{Citation
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| | last=Belabas
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| | first=Karim
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| | title=A fast algorithm to compute cubic fields
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| | year=1997
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| | journal=Mathematics of Computation
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| | volume=66
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| | number=219
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| | pages=1213–1237
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| | mr=1415795
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| }}
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| *{{cite arxiv
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| | last=Bhargava
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| | first=Manjul
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| | author-link=Manjul Bhargava
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| | last2=Shankar
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| | first2=Arul
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| | last3=Tsimerman
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| | first3=Jacob
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| | title=On the Davenport–Heilbronn theorem and second order terms
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| | year=2010
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| | eprint=1005.0672 }}
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| *{{Citation
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| | last=Cohen
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| | first=Henri
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| | author-link=Henri Cohen (number theorist)
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| | title=A Course in Computational Algebraic Number Theory
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| | publisher=[[Springer-Verlag]]
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| | location=Berlin, New York
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| | series=Graduate Texts in Mathematics
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| | isbn=978-3-540-55640-4
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| | mr=1228206
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| | year=1993
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| | volume=138
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| }}
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| *{{Citation
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| | last=Cohn
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| | first=Harvey
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| | title=The density of abelian cubic fields
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| | year=1954
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| | journal=[[Proceedings of the American Mathematical Society]]
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| | volume=5
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| | pages=476–477
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| | mr=0064076
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| }}
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| *{{Citation
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| | last=Davenport
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| | first=Harold
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| | author-link=Harold Davenport
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| | last2=Heilbronn
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| | first2=Hans
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| | author2-link=Hans Heilbronn
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| | title=On the density of discriminants of cubic fields. II
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| | year=1971
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| | journal=[[Proceedings of the Royal Society A]]
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| | volume=322
| |
| | number=1551
| |
| | pages=405–420
| |
| | mr=0491593
| |
| }}
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| *{{Citation
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| | last=Hasse
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| | first=Helmut
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| | author-link=Helmut Hasse
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| | title=Arithmetische Theorie der kubischen Zahlkörper auf klassenkörpertheoretischer Grundlage
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| | year=1930
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| | language=German
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| | journal=Mathematische Zeitschrift
| |
| | pages=565–582
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| | volume=31
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| | number=1
| |
| | doi=10.1007/BF01246435
| |
| }}
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| *{{Citation
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| | last=Roberts
| |
| | first=David P.
| |
| | title=Density of cubic field discriminants
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| | year=2001
| |
| | journal=Mathematics of Computation
| |
| | volume=70
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| | number=236
| |
| | pages=1699–1705
| |
| | mr=1836927
| |
| }}
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| *{{cite arxiv
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| | last=Taniguchi
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| | first=Takashi
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| | last2=Throne
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| | first2=Frank
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| | title=Secondary terms in counting functions for cubic fields
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| | year=2011
| |
| | eprint=1102.2914 }}
| |
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| [[Category:Algebraic number theory]]
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| [[Category:Field theory]]
| |
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