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{{Short description|In number theory, measure of non-unique factorization}}
{{more citations needed|date=February 2010}}
In [[mathematics]], the '''ideal class group''' (or '''class group''') of an  [[algebraic number field]] {{math|''K''}} is the [[quotient group]] {{math|''J<sub>K</sub>''&hairsp;/''P<sub>K</sub>''}} where {{math|''J<sub>K</sub>''}} is the [[group (mathematics)|group]] of [[fractional ideal]]s of the [[ring of integers]] of {{math|''K''}}, and {{math|''P<sub>K</sub>''}} is its [[subgroup]] of [[principal fractional ideal|principal]] ideals. The class group is a measure of the extent to which [[unique factorization]] fails in the ring of integers of {{math|''K''}}. The [[order of a group|order]] of the group, which is [[finite group|finite]], is called the '''class number''' of {{math|''K''}}.

The theory extends to [[Dedekind domain]]s and their [[field of fractions|fields of fractions]], for which the multiplicative properties are intimately tied to the structure of the class group. For example, the class group of a Dedekind domain is [[trivial group|trivial]] [[if and only if]] the ring is a [[unique factorization domain]].

== History and origin of the ideal class group ==

Ideal class groups (or, rather, what were effectively ideal class groups) were studied some time before the idea of an [[Ideal (ring theory)|ideal]] was formulated. These groups appeared in the theory of [[quadratic form]]s: in the case of [[binary quadratic form|binary]] integral quadratic forms, as put into something like a final form by [[Carl Friedrich Gauss]], a composition law was defined on certain [[equivalence class]]es of forms. This gave a finite [[abelian group]], as was recognised at the time.

Later [[Ernst Kummer]] was working towards a theory of [[cyclotomic field]]s. It had been realised (probably by several people) that failure to complete [[mathematical proof|proofs]] in the general case of [[Fermat's Last Theorem]] by factorisation using the [[roots of unity]] was for a very good reason: a failure of unique factorization – i.e., the [[fundamental theorem of arithmetic]] – to hold in the [[ring (mathematics)|ring]]s generated by those roots of unity was a major obstacle. Out of Kummer's work for the first time came a study of the obstruction to the factorization. We now recognise this as part of the ideal class group: in fact Kummer had isolated the ''p''-[[torsion subgroup|torsion]] in that group for the [[field (mathematics)|field]] of ''p''-roots of unity, for any [[prime number]] ''p'', as the reason for the failure of the standard method of attack on the Fermat problem (see [[regular prime]]).

Somewhat later again [[Richard Dedekind]] formulated the concept of an ideal, Kummer having worked in a different way. At this point the existing examples could be unified. It was shown that while rings of [[algebraic integer]]s do not always have unique factorization into primes (because they need not be [[principal ideal domain]]s), they do have the property that every [[proper ideal]] admits a unique factorization as a product of [[prime ideal]]s (that is, every ring of algebraic integers is a [[Dedekind domain]]). The size of the ideal class group can be considered as a measure for the deviation of a ring from being a principal ideal domain; a ring is a principal ideal domain if and only if it has a trivial ideal class group.

== Definition ==

If ''R'' is an [[integral domain]], define a [[relation (mathematics)|relation]] ~ on nonzero [[fractional ideal]]s of ''R'' by ''I'' ~ ''J'' whenever there exist nonzero elements ''a'' and ''b'' of ''R'' such that (''a'')''I'' = (''b'')''J''. (Here the notation (''a'') means the [[principal ideal]] of ''R'' consisting of all the multiples of ''a''.) It is easily shown that this is an [[equivalence relation]]. The equivalence classes are called the ''ideal classes'' of ''R''.
Ideal classes can be multiplied: if [''I''] denotes the equivalence class of the ideal ''I'', then the multiplication [''I''][''J''] = [''IJ''] is well-defined and [[commutative]]. The principal ideals form the ideal class [''R''] which serves as an [[identity element]] for this multiplication. Thus a class [''I''] has an [[inverse element|inverse]] [''J''] if and only if there is an ideal ''J'' such that ''IJ'' is a principal ideal. In general, such a ''J'' may not exist and consequently the set of ideal classes of ''R'' may only be a [[monoid]].

However, if ''R'' is the ring of [[algebraic integer]]s in an [[algebraic number field]], or more generally a [[Dedekind domain]], the multiplication defined above turns the set of fractional ideal classes into an [[abelian group]], the '''ideal class group''' of ''R''. The group property of existence of inverse elements follows easily from the fact that, in a Dedekind domain, every non-zero ideal (except ''R'') is a product of [[prime ideal]]s.

==Properties<!--'Class number (number theory)' redirects here-->==
The ideal class group is trivial (i.e. has only one element) if and only if all ideals of ''R'' are principal. In this sense, the ideal class group measures how far ''R'' is from being a [[principal ideal domain]], and hence from satisfying unique prime factorization (Dedekind domains are [[unique factorization domain]]s if and only if they are principal ideal domains).

The number of ideal classes (the '''{{vanchor|class number}}'''<!--boldface per WP:R#PLA--> of ''R'') may be infinite in general. In fact, every abelian group is [[group isomorphism|isomorphic]] to the ideal class group of some Dedekind domain.<ref>{{harvnb|Claborn|1966}}</ref> But if ''R'' is a ring of algebraic integers, then the class number is always ''finite''. This is one of the main results of classical [[algebraic number theory]].

Computation of the class group is hard, in general; it can be done by hand for the ring of integers in an [[algebraic number field]] of small [[discriminant of an algebraic number field|discriminant]], using [[Minkowski's bound]]. This result gives a bound, depending on the ring, such that every ideal class contains an [[ideal norm]] less than the bound. In general the bound is not sharp enough to make the calculation practical for fields with large discriminant, but computers are well suited to the task.

The mapping from rings of integers ''R'' to their corresponding class groups is [[functor]]ial, and the class group can be subsumed under the heading of [[algebraic K-theory]], with ''K''<sub>0</sub>(''R'') being the functor assigning to ''R'' its ideal class group; more precisely, ''K''<sub>0</sub>(''R'') = '''Z'''×''C''(''R''), where ''C''(''R'') is the class group. Higher K groups can also be employed and interpreted arithmetically in connection to rings of integers.

==Relation with the group of units==
It was remarked above that the ideal class group provides part of the answer to the question of how much ideals in a [[Dedekind domain]] behave like elements. The other part of the answer is provided by the [[group of units]] of the Dedekind domain, since passage from principal ideals to their generators requires the use of [[unit (ring theory)|units]] (and this is the rest of the reason for introducing the concept of fractional ideal, as well):

Define a map from ''R''<sup>×</sup> to the set of all nonzero fractional ideals of ''R'' by sending every element to the principal (fractional) ideal it generates. This is a [[group homomorphism]]; its [[Kernel (algebra)|kernel]] is the group of units of ''R'', and its [[cokernel]] is the ideal class group of ''R''. The failure of these groups to be trivial is a measure of the failure of the map to be an isomorphism: that is the failure of ideals to act like ring elements, that is to say, like numbers.

== Examples of ideal class groups ==

* The rings '''[[Integer#Algebraic properties|Z]]''', [[Eisenstein integer|'''Z'''[ω]]], and [[Gaussian integer|'''Z'''[''i'']]], where ω is a cube [[root of unity|root of 1]] and ''i'' is a fourth root of 1 (i.e. a [[imaginary unit|square root of −1]]), are all principal ideal domains (and in fact are all [[Euclidean domain]]s), and so have class number 1: that is, they have trivial ideal class groups. 
*If ''k'' is a field, then the [[polynomial ring]] ''k''[''X''<sub>1</sub>, ''X''<sub>2</sub>, ''X''<sub>3</sub>, ...] is an integral domain. It has a countably infinite set of ideal classes.

=== Class numbers of quadratic fields ===
If <math>d</math> is a [[square-free integer]] (a product of distinct primes) other than 1, then <math>\mathbf{Q}(\sqrt{d}\,)</math> is a [[quadratic field|quadratic extension of '''Q''']].  If <math>d < 0</math>, then the class number of the ring <math>R</math> of algebraic integers of <math>\mathbf{Q}(\sqrt{d}\,)</math> is equal to 1 for precisely the following values of <math>d</math>: <math>d = -1,-2,-3,-7,-11, -19, -43, -67, -163</math>. This result was first [[conjecture]]d by [[Carl Friedrich Gauss|Gauss]] and proven by [[Kurt Heegner]], although Heegner's proof was not believed until [[Harold Stark]] gave a later proof in 1967. (See [[Stark–Heegner theorem]].) This is a special case of the famous [[class number problem]].

If, on the other hand, ''d'' > 0, then it is unknown whether there are infinitely many fields <math>\mathbf{Q}(\sqrt{d}\,)</math> with class number 1. Computational results indicate that there are a great many such fields. However, it is not even known if there are infinitely many [[number field]]s with class number 1.<ref>{{harvnb|Neukirch|1999}}</ref>

For ''d'' < 0, the ideal class group of <math>\mathbf{Q}(\sqrt{d}\,)</math> is isomorphic to the class group of integral [[binary quadratic form]]s of [[discriminant of a quadratic form|discriminant]] equal to the discriminant of <math>\mathbf{Q}(\sqrt{d}\,)</math>. For ''d'' > 0, the ideal class group may be half the size since the class group of integral binary quadratic forms is isomorphic to the [[narrow class group]] of <math>\mathbf{Q}(\sqrt{d}\,)</math>.<ref>{{harvnb|Fröhlich|Taylor|1993|loc=Theorem 58}}</ref>

For [[real number|real]] [[quadratic integer]] rings, the class number is given in [https://oeis.org/A003649 OEIS A003649]; for the [[imaginary number|imaginary]] case, they are given in [https://oeis.org/A000924 OEIS A000924].

==== Example of a non-trivial class group ====

The quadratic integer ring ''R'' = '''Z'''[{{radic|−5}}] is the ring of integers of '''Q'''({{radic|−5}}). It does ''not'' possess unique factorization; in fact the class group of ''R'' is [[cyclic group|cyclic]] of order 2. Indeed, the ideal
: ''J'' = (2, 1 + {{radic|−5}})
is not principal, which can be [[proof by contradiction|proved by contradiction]] as follows. <math>R</math> has a [[field norm|norm]] function <math>N(a + b \sqrt{-5}) =  a^2 + 5 b^2 </math>, which satisfies <math>N(uv) = N(u)N(v)</math>, and <math>N(u) = 1</math> if and only if <math>u</math> is a unit in <math>R</math>. First of all, <math> J \ne R</math>, because the [[quotient ring]] of <math>R</math> modulo the ideal <math>(1 + \sqrt{-5})</math> is [[ring isomorphism|isomorphic]] to <math>\mathbf{Z} / 6 \mathbf{Z}</math>, so that the quotient ring of <math>R</math> modulo <math>J</math> is isomorphic to <math>\mathbf{Z} / 3 \mathbf{Z}</math>.  If ''J'' were generated by an element ''x'' of ''R'', then ''x'' would divide both 2 and 1 + {{radic|−5}}. Then the norm <math>N(x)</math> would divide both <math>N(2) = 4</math> and <math>N(1 + \sqrt{-5}) = 6</math>, so ''N''(x) would divide 2. If <math>N(x) = 1</math> then <math>x</math> is a unit and <math>J = R</math>, a contradiction. But <math>N(x)</math> cannot be 2 either, because ''R'' has no elements of norm 2, because the [[Diophantine equation]] <math>b^2 + 5 c^2 = 2</math> has no solutions in integers, as it has no solutions [[modular arithmetic|modulo 5]].

One also computes that ''J''<sup>&thinsp;2</sup> = (2), which is principal, so the class of ''J'' in the ideal class group has order two. Showing that there aren't any ''other'' ideal classes requires more effort. <!-- well, I can't think of any method apart from the minkowski bound off the top of my head -->

The fact that this ''J'' is not principal is also related to the fact that the element 6 has two distinct factorisations into [[irreducible element|irreducibles]]:
: 6 = 2 × 3 = (1 + {{radic|−5}}) × (1 − {{radic|−5}}).

== Connections to class field theory ==

[[Class field theory]] is a branch of [[algebraic number theory]] which seeks to classify all the [[abelian extension]]s of a given algebraic number field, meaning [[Galois extension]]s with abelian [[Galois group]]. A particularly beautiful example is found in the [[Hilbert class field]] of a number field, which can be defined as the maximal [[unramified]] abelian extension of such a field. The Hilbert class field ''L'' of a number field ''K'' is unique and has the following properties:

* Every ideal of the ring of integers of ''K'' becomes principal in ''L'', i.e., if ''I'' is an integral ideal of ''K'' then the image of ''I'' is a principal ideal in ''L''.
* ''L'' is a Galois extension of ''K'' with Galois group isomorphic to the ideal class group of ''K''.

Neither property is particularly easy to prove.

== See also ==
* [[Class number formula]]
* [[Class number problem]]
* [[Brauer–Siegel theorem]]—an [[Asymptotic analysis|asymptotic]] formula for the class number
* [[List of number fields with class number one]]
* [[Principal ideal domain]]
* [[Algebraic K-theory]]
* [[Galois theory]]
* [[Fermat's Last Theorem]]
* [[Narrow class group]]
* [[Picard group]]—a generalisation of the class group appearing in [[algebraic geometry]]
* [[Arakelov class group]]

== Notes ==
{{Reflist}}

== References ==
*{{Citation
 |first       = Luther
 |last        = Claborn
 |title       = Every abelian group is a class group
 |journal     = Pacific Journal of Mathematics
 |volume      = 18
 |year        = 1966
 |issue = 2
 |pages       = 219–222
 |doi         = 10.2140/pjm.1966.18.219
 |doi-access= free
 }}
* {{Citation
| last1= Fröhlich
| first1 = Albrecht
| authorlink= Albrecht Fröhlich
| last2=Taylor
| first2=Martin
| authorlink2= Martin J. Taylor
| title=Algebraic number theory
| publisher=[[Cambridge University Press]]
| series=Cambridge Studies in Advanced Mathematics
| isbn=978-0-521-43834-6
| mr=1215934
| year=1993
| volume=27
}}
*{{Neukirch ANT}}

[[Category:Algebraic number theory]]
[[Category:Ideals (ring theory)]]