View source for Discriminant of an algebraic number field

{{short description|Measures the size of the ring of integers of the algebraic number field}}
{{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''<sup>3</sup>&nbsp;&minus;&nbsp;''x''<sup>2</sup>&nbsp;&minus;&nbsp;2''x''&nbsp;+&nbsp;1. This fundamental domain sits inside ''K''&nbsp;&otimes;<sub>'''Q'''</sub>&nbsp;'''R'''. The discriminant of ''K'' is 49&nbsp;=&nbsp;7<sup>2</sup>. Accordingly, the volume of the fundamental domain is 7 and ''K'' is only [[Ramification (mathematics)|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 proportional to the squared 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''. [[Hermite–Minkowski theorem|A 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''&nbsp;=&nbsp;'''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''<sub>''K''</sub> be its [[ring of integers]]. Let ''b''<sub>1</sub>, ..., ''b''<sub>''n''</sub> be an [[integral basis]] of ''O''<sub>''K''</sub> (i.e. a basis as a [[Module (mathematics)|'''Z'''-module]]), and let {σ<sub>1</sub>, ..., σ<sub>''n''</sub>} be the set of embeddings of ''K'' into the [[complex number]]s (i.e. [[injective]] [[ring homomorphism]]s ''K''&nbsp;→&nbsp;'''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<sub>''i''</sub>(''b''<sub>''j''</sub>). Symbolically,
:<math>\Delta_K=\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)^2.
</math>
<br>
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'''<sub>''K''/'''Q'''</sub>(''b''<sub>''i''</sub>''b''<sub>''j''</sub>). This matrix equals ''B''<sup>T</sup>''B'', so the square of the discriminant of ''K'' is the determinant of this matrix.

The discriminant of an [[Order_(ring_theory)|order]] in K with [[integral basis]] ''b''<sub>1</sub>, ..., ''b''<sub>''n''</sub> is defined in the same way.

==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''&nbsp;>&nbsp;2 be an integer, let ζ<sub>''n''</sub> be a [[Root of unity|primitive ''n''th root of unity]], and let ''K''<sub>''n''</sub> = '''Q'''(ζ<sub>''n''</sub>) be the ''n''th cyclotomic field. The discriminant of ''K''<sub>''n''</sub> is given by<ref name=MP130>{{citation | first1=Yu. I. | last1=Manin | author-link1=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><ref>Proposition 2.7 of {{harvnb|Washington|1997}}</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''<sub>''K''</sub> = '''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 choose the integral basis of ''O''<sub>''K''</sub> to be ''b''<sub>1</sub>&nbsp;=&nbsp;1, ''b''<sub>2</sub>&nbsp;=&nbsp;α, ''b''<sub>3</sub>&nbsp;=&nbsp;''α''<sup>2</sup>, ..., ''b''<sub>''n''</sub>&nbsp;=&nbsp;''α''<sup>''n''&minus;1</sup>. Then, the matrix in the definition is the [[Vandermonde matrix]] associated to α<sub>''i''</sub>&nbsp;=&nbsp;σ<sub>''i''</sub>(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''<sup>3</sup>&nbsp;&minus;&nbsp;''x''<sup>2</sup>&nbsp;&minus;&nbsp;2''x''&nbsp;&minus;&nbsp;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&nbsp;+&nbsp;1)/2} and the discriminant of ''K'' is &minus;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''<sup>3</sup> &minus; 21''x'' + 28}} or {{nobreak|''x''<sup>3</sup> &minus; 21''x'' &minus; 35}}, respectively.<ref>{{harvnb|Cohen|1993|loc=Theorem 6.4.6}}</ref>
<!-- Non-power basis example lacking reference
*Let ''K'' = '''Q'''(α) be the number field obtained by adjoining a root α of the polynomial ''x''<sup>3</sup>&nbsp;&minus;&nbsp;11''x''<sup>2</sup>&nbsp;+&nbsp;''x''&nbsp;+&nbsp;1. This is an example that does not have a power basis. An integral basis is given by {1, α, 1/2(α<sup>2</sup>&nbsp;+&nbsp;1)}, and the trace form is
:: <math>\left(\begin{array}{ccc}
3 & 11 & 61 \\
11 & 119 & 653 \\
61 & 653 & 3589 \\
\end{array}\right).</math>
: The discriminant of ''K'' is the determinant of this matrix, which is 1304 = 2<sup>3</sup> 163.
-->

==Basic results==
*'''Brill's theorem''':<ref>{{harvnb|Koch|1997|p=11}}</ref> The [[sign (mathematics)|sign]] of the discriminant is (&minus;1)<sup>''r''<sub>2</sub></sup> where ''r''<sub>2</sub> 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<sub>''K''</sub>&nbsp;.<ref>Corollary III.2.12 of {{harvnb|Neukirch|1999}}</ref><ref>{{cite web|title=Discriminants and ramified primes|last=Conrad|first=Keith|url=https://kconrad.math.uconn.edu/blurbs/gradnumthy/disc.pdf#theorem.1.3|quote=Theorem 1.3 (Dedekind). For a number field K, a prime p ramifies in K if and only if p divides the integer discZ(OK)}}</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''<sub>2</sub> 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<sub>''K''</sub>| > 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<sub>''K''</sub>| &lt; ''N''.  Again, this follows from the Minkowski bound together with Hermite's theorem (that 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<sub>''K''/''L''</sub> of an extension of number fields ''K''/''L'', which is an ideal in ''O''<sub>''L''</sub>. The relative discriminant is defined in a fashion similar to the absolute discriminant, but must take into account that ideals in ''O''<sub>''L''</sub> may not be principal and that there may not be an ''O''<sub>''L''</sub> basis of ''O''<sub>''K''</sub>. Let {σ<sub>1</sub>, ..., σ<sub>''n''</sub>} be the set of embeddings of ''K'' into '''C''' which are the identity on ''L''. If ''b''<sub>1</sub>, ..., ''b''<sub>''n''</sub> is any basis of ''K'' over ''L'', let ''d''(''b''<sub>1</sub>, ..., ''b''<sub>''n''</sub>) be the square of the determinant of the ''n'' by ''n'' matrix whose (''i'',''j'')-entry is σ<sub>''i''</sub>(''b''<sub>''j''</sub>). Then, the relative discriminant of ''K''/''L'' is the ideal generated by the ''d''(''b''<sub>1</sub>, ..., ''b''<sub>''n''</sub>) as {''b''<sub>1</sub>, ..., ''b''<sub>''n''</sub>} varies over all integral bases of ''K''/''L''. (i.e. bases with the property that ''b<sub>i</sub>''&nbsp;∈&nbsp;''O<sub>K</sub>'' 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<sub>''K''/'''Q'''</sub> is the principal ideal of '''Z''' generated by the absolute discriminant Δ<sub>''K''</sub>&nbsp;. 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<sub>''K''/''L''</sub>. 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 degree ''n'' number field ''K'' is defined by the formula

:<math>\operatorname{rd}_K = |\Delta_K|^{1/n}.</math><ref name=HajirMaire>{{cite journal| last1=Hajir | first1=Farshid | last2=Maire | first2=Christian | title=Tamely ramified towers and discriminant bounds for number fields. II | journal=[[Journal of Symbolic Computation|J. Symbolic Comput.]] | volume=33 | pages=415–423 | year=2002 | doi=10.1023/A:1017537415688 | doi-access=free }}</ref>

The relation between relative discriminants in a tower of fields shows that the root discriminant does not change in an unramified extension.
  
===Asymptotic lower bounds===
Given nonnegative rational numbers ''ρ'' and ''σ'', not both 0, and a positive integer ''n'' such that the pair (''r'',2''s'')&nbsp;=&nbsp;(''ρn'',''σn'') is in '''Z'''&nbsp;×&nbsp;2'''Z''', let ''α''<sub>''n''</sub>(''ρ'',&nbsp;''σ'') be the infimum of rd<sub>''K''</sub> as ''K'' ranges over degree ''n'' number fields with ''r'' real embeddings and 2''s'' complex embeddings, and let ''α''(''ρ'',&nbsp;''σ'')&nbsp;=&nbsp; liminf<sub>''n''→∞&nbsp;</sub>''α''<sub>''n''</sub>(''ρ'',&nbsp;''σ'').  Then

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

and the [[generalized Riemann hypothesis]] implies the stronger bound

:<math> \alpha(\rho,\sigma) \ge 215.3^\rho 44.7^\sigma .</math><ref name=koch181>{{harvnb|Koch|1997|pp=181–182}}</ref>

There is also a lower bound that holds in all degrees, not just asymptotically: For totally real fields, the root discriminant is >&nbsp;14, with 1229 exceptions.<ref name=voight2008>{{harvnb|Voight|2008}}</ref>

===Asymptotic upper bounds===
On the other hand, the existence of an infinite [[class field tower]] can give upper bounds on the values of ''α''(''ρ'',&nbsp;''σ'').  For example, the infinite class field tower over '''Q'''({{radic|-''m''}}) with ''m''&nbsp;=&nbsp;3·5·7·11·19 produces fields of arbitrarily large degree with root discriminant 2{{radic|''m''}} ≈ 296.276,<ref name=koch181/> so ''α''(0,1)&nbsp;<&nbsp;296.276.  Using ''tamely ramified'' towers, Hajir and Maire have shown that ''α''(1,0)&nbsp;<&nbsp;954.3 and ''α''(0,1)&nbsp;<&nbsp;82.2,<ref name=HajirMaire/> improving upon earlier bounds of Martinet.<ref name=koch181/><ref>{{cite journal | zbl=0369.12007 | last=Martinet | first=Jacques | title=Tours de corps de classes et estimations de discriminants | language=fr | journal=[[Inventiones Mathematicae]] | volume=44 | pages=65–73 | year=1978 | doi=10.1007/bf01389902| bibcode=1978InMat..44...65M | s2cid=122278145 }}</ref>  

==Relation to other quantities==
*When embedded into <math>K\otimes_\mathbf{Q}\mathbf{R}</math>, the volume of the fundamental domain of ''O''<sub>''K''</sub> 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''<sub>2</sub> 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==

===Primary sources===
* {{Citation
| last=Brill
| first=Alexander von
| title=Ueber die Discriminante
| year=1877
| journal=Mathematische Annalen
| volume=12
| pages=87–89
| doi=10.1007/BF01442468
| url=http://gdz.sub.uni-goettingen.de/en/dms/load/toc/?PPN=PPN235181684_0012&DMDID=dmdlog8
| access-date=2009-08-22
| issue=1
| mr=1509928
| jfm=09.0059.02
| s2cid=120947279
}}
* {{Citation
| last=Dedekind
| first=Richard
| author-link=Richard Dedekind
| title=Vorlesungen über Zahlentheorie von P.G. Lejeune Dirichlet
| url=http://gdz.sub.uni-goettingen.de/en/dms/load/toc/?PPN=PPN30976923X&DMDID=dmdlog1
| edition=2
| year=1871
| publisher=Vieweg
| access-date=2009-08-05
}}
* {{Citation
| last=Dedekind
| first=Richard
| author-link=Richard Dedekind
| title=Über den Zusammenhang zwischen der Theorie der Ideale und der Theorie der höheren Congruenzen
| url=http://gdz.sub.uni-goettingen.de/en/dms/load/toc/?PPN=PPN250442582_0023&DMDID=dmdlog10
| year=1878
| journal=Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen
| volume=23
| access-date=2009-08-20
| issue=1
}}
*{{Citation
| last=Hermite
| first=Charles
| author-link=Charles Hermite
| title=Extrait d'une lettre de M. C. Hermite à M. Borchardt sur le nombre limité d'irrationalités auxquelles se réduisent les racines des équations à coefficients entiers complexes d'un degré et d'un discriminant donnés
| journal=[[Crelle's Journal]]
| volume=1857
| issue=53
| pages=182–192
| year=1857
| url=http://gdz.sub.uni-goettingen.de/dms/load/toc/?PPN=PPN243919689_0053&DMDID=dmdlog14
| access-date=2009-08-20
| doi=10.1515/crll.1857.53.182
| s2cid=120694650
}}
*{{Citation
| last=Kronecker
| first=Leopold
| author-link=Leopold Kronecker
| title=Grundzüge einer arithmetischen Theorie der algebraischen Grössen
| url=http://gdz.sub.uni-goettingen.de/dms/load/toc/?PPN=PPN243919689_0092&DMDID=dmdlog4
| journal=[[Crelle's Journal]]
| volume=92
| year=1882
| pages=1–122
| access-date=2009-08-20
| jfm=14.0038.02
}}
*{{Citation
| last=Minkowski
| first=Hermann
| author-link=Hermann Minkowski
| title=Ueber die positiven quadratischen Formen und über kettenbruchähnliche Algorithmen
| url=http://gdz.sub.uni-goettingen.de/dms/load/toc/?PPN=PPN243919689_0107&DMDID=dmdlog25
| journal=Crelle's Journal
| volume=1891
| issue=107
| year=1891a
| pages=278–297
| access-date=2009-08-20
| jfm=23.0212.01
| doi=10.1515/crll.1891.107.278
}}
*{{Citation
| last=Minkowski
| first=Hermann
| author-link=Hermann Minkowski
| title=Théorèmes d'arithmétiques
| journal=[[Comptes rendus de l'Académie des sciences]]
| year=1891b
| volume=112
| pages=209–212
| jfm=23.0214.01
}}
*{{Citation
| last=Stickelberger
| first=Ludwig
| author-link=Ludwig Stickelberger
| contribution=Über eine neue Eigenschaft der Diskriminanten algebraischer Zahlkörper
| title=Proceedings of the First International Congress of Mathematicians, Zürich
| year=1897
| pages=182–193
| jfm=29.0172.03
}}

===Secondary sources===
* {{Bourbaki EHM}}
* {{Citation
| last=Cohen
| first=Henri
| author-link=Henri Cohen (number theorist)
| title=A Course in Computational Algebraic Number Theory
| publisher=[[Springer-Verlag]]
| location=Berlin, New York
| series=Graduate Texts in Mathematics
| isbn=978-3-540-55640-4
| year=1993
| volume=138
| mr=1228206
}}
* {{Citation
| last1=Cohen
| first1=Henri
| author-link=Henri Cohen (number theorist)
| last2=Diaz y Diaz
| first2=Francisco
| last3=Olivier
| first3=Michel
| contribution=A Survey of Discriminant Counting
| title=Algorithmic Number Theory, Proceedings, 5th International Syposium, ANTS-V, University of Sydney, July 2002
| editor-last=Fieker
| editor-first=Claus
| editor2-last=Kohel
| editor2-first=David R.
| publisher=Springer-Verlag
| location=Berlin
| series=Lecture Notes in Computer Science
| issn=0302-9743
| isbn=978-3-540-43863-2
| doi=10.1007/3-540-45455-1_7
| year=2002
| volume=2369
| pages=80–94
| mr=2041075
}}
* {{Citation
| last1= Fröhlich
| first1 = Albrecht
| author-link= Albrecht Fröhlich
| last2=Taylor
| first2=Martin
| author-link2= Martin J. Taylor
| title=Algebraic number theory
| publisher=[[Cambridge University Press]]
| series=Cambridge Studies in Advanced Mathematics
| isbn=978-0-521-43834-6
| year=1993
| volume=27
| mr= 1215934
}}
* {{citation | 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}}
* {{Citation
| last=Narkiewicz
| first=Władysław
| title=Elementary and analytic theory of algebraic numbers
| edition=3
| year=2004
| publisher=Springer-Verlag
| location=Berlin
| series=Springer Monographs in Mathematics
| isbn=978-3-540-21902-6
| mr=2078267
}}
* {{Neukirch ANT}}
* {{Citation
| last=Serre
| first=Jean-Pierre
| author-link=Jean-Pierre Serre
| chapter=Local class field theory
| title=Algebraic Number Theory, Proceedings of an instructional conference at the University of Sussex, Brighton, 1965
| editor-last=Cassels
| editor-first=J. W. S.
| editor-link=J. W. S. Cassels
| editor2-last=Fröhlich
| editor2-first=Albrecht
| editor2-link=Albrecht Fröhlich
| publisher=Academic Press
| location=London
| isbn=0-12-163251-2
| year=1967
| mr=0220701
}}
* {{Citation
| first = John
| last = Voight
| contribution = Enumeration of totally real number fields of bounded root discriminant
| title = Algorithmic number theory. Proceedings, 8th International Symposium, ANTS-VIII, Banff, Canada, May 2008
| editor1-last=van der Poorten | editor1-first=Alfred J. | editor1-link=Alfred van der Poorten
| editor2-last=Stein
| editor2-first=Andreas
| pages = 268–281
| publisher = Springer-Verlag
| location=Berlin
| series=Lecture Notes in Computer Science
| volume=5011
| year = 2008
| arxiv = 0802.0194
| doi = 10.1007/978-3-540-79456-1_18
| isbn=978-3-540-79455-4| mr = 2467853 | zbl=1205.11125 
| s2cid = 30036220
}}
* {{Citation
| last=Washington
| first=Lawrence
| title=Introduction to Cyclotomic Fields
| edition=2 nd
| publisher=Springer-Verlag
| location=Berlin, New York
| series=Graduate Texts in Mathematics
| isbn=978-0-387-94762-4
| year=1997
| volume=83
| mr=1421575 | zbl=0966.11047
}}

==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 | access-date=2008-08-20}}

[[Category:Algebraic number theory]]