Ideal class group
Encyclopedia
In mathematics
, the extent to which unique factorization fails in the ring of integers
of an algebraic number field
(or more generally any Dedekind domain
) can be described by a certain group
known as an ideal class group (or class group). If this group is finite (as it is in the case of the ring of integers of a number field), then the order
of the group is called the class number.
The multiplicative theory of a Dedekind domain is intimately tied to the structure of its class group. For example, the class group of a Dedekind domain is trivial if and only if the ring is a unique factorization domain
.
was formulated. These groups appeared in the theory of quadratic form
s: in the case of binary integral quadratic forms, as put into something like a final form by Gauss
, a composition law was defined on certain equivalence classes of forms. This gave a finite abelian group
, as was recognised at the time.
Later Kummer
was working towards a theory of cyclotomic field
s. It had been realised (probably by several people) that failure to complete 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 the fundamental theorem of arithmetic
to hold in the 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 factorisation. We now recognise this as part of the ideal class group: in fact Kummer had isolated the p-torsion
in that group for the 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 Dedekind
formulated the concept of 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 domain; a ring is a principal domain if and only if it has a trivial ideal class group.
~ 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 [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 integers 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 element
s follows easily from the fact that, in a Dedekind domain, every non-zero ideal (except R) is a product of prime ideals.
, 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 class number of R) may be infinite in general. In fact, every abelian group is isomorphic to the ideal class group of some Dedekind domain. But if R is in fact 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
, using Minkowski's bound
. This result gives a bound, depending on the ring, such that every ideal class contains an ideal of 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 functorial, and the class group can be subsumed under the heading of algebraic K-theory
, with K0(R) being the functor assigning to R its ideal class group; more precisely, K0(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.
behave like elements. The other part of the answer is provided by the multiplicative group
of unit
s of the Dedekind domain, since passage from principal ideals
to their generators requires the use of units (and this is the rest of the reason for introducing the concept of fractional ideal, as well):
Define a map from K× 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
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.
(a product of distinct primes) other than 1, then Q(√d) is a quadratic extension of Q
. If d < 0, then the class number of the ring R of algebraic integers of Q(√d) is equal to 1 for precisely the following values of d: d = −1, −2, −3, −7, −11, −19, −43, −67, and −163. This result was first conjectured by 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 Q(√d) 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 fields with class number 1.
For d < 0, the ideal class group of Q(√d) is isomorphic to the class group of integral binary quadratic form
s of discriminant
equal to the discriminant of Q(√d). 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 Q(√d).
ring R = Z [√−5] is the ring of integers of Q(√−5). It does not possess unique factorization; in fact the class group of R is cyclic of order 2. Indeed, the ideal
is not principal, which can be proved by contradiction as follows. If J were generated by an element x of R, then x would divide both 2 and 1 + √−5. Then the norm
N(x) of x would divide both N(2) = 4 and N(1 + √−5) = 6, so N(x) would divide 2. We are assuming that x is not a unit of R, so N(x) cannot be 1. It cannot be 2 either, because R has no elements of norm 2, that is, the equation b2 + 5c2 = 2 has no solutions in integers.
One also computes that J2 = (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.
The fact that this J is not principal is also related to the fact that the element 6 has two distinct factorisations into irreducibles:
is a branch of algebraic number theory
which seeks to classify all the abelian extension
s of a given algebraic number field, meaning Galois extensions 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:
Neither property is particularly easy to prove.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the extent to which unique factorization fails in the ring of integers
Ring of integers
In mathematics, the ring of integers is the set of integers making an algebraic structure Z with the operations of integer addition, negation, and multiplication...
of an algebraic number field
Algebraic number field
In mathematics, an algebraic number field F is a finite field extension of the field of rational numbers Q...
(or more generally any Dedekind domain
Dedekind domain
In abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors into a product of prime ideals. It can be shown that such a factorization is then necessarily unique up to the order of the factors...
) can be described by a certain group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
known as an ideal class group (or class group). If this group is finite (as it is in the case of the ring of integers of a number field), then the order
Order (group theory)
In group theory, a branch of mathematics, the term order is used in two closely related senses:* The order of a group is its cardinality, i.e., the number of its elements....
of the group is called the class number.
The multiplicative theory of a Dedekind domain is intimately tied to the structure of its class group. For example, the class group of a Dedekind domain is trivial if and only if the ring is a unique factorization domain
Unique factorization domain
In mathematics, a unique factorization domain is, roughly speaking, a commutative ring in which every element, with special exceptions, can be uniquely written as a product of prime elements , analogous to the fundamental theorem of arithmetic for the integers...
.
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 idealIdeal (ring theory)
In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. The ideal concept allows the generalization in an appropriate way of some important properties of integers like "even number" or "multiple of 3"....
was formulated. These groups appeared in the theory of quadratic form
Quadratic form
In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. For example,4x^2 + 2xy - 3y^2\,\!is a quadratic form in the variables x and y....
s: in the case of binary integral quadratic forms, as put into something like a final form by Gauss
Carl Friedrich Gauss
Johann Carl Friedrich Gauss was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy and optics.Sometimes referred to as the Princeps mathematicorum...
, a composition law was defined on certain equivalence classes of forms. This gave a finite abelian group
Abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...
, as was recognised at the time.
Later Kummer
Ernst Kummer
Ernst Eduard Kummer was a German mathematician. Skilled in applied mathematics, Kummer trained German army officers in ballistics; afterwards, he taught for 10 years in a gymnasium, the German equivalent of high school, where he inspired the mathematical career of Leopold Kronecker.-Life:Kummer...
was working towards a theory of cyclotomic field
Cyclotomic field
In number theory, a cyclotomic field is a number field obtained by adjoining a complex primitive root of unity to Q, the field of rational numbers...
s. It had been realised (probably by several people) that failure to complete proofs in the general case of Fermat's last theorem
Fermat's Last Theorem
In number theory, Fermat's Last Theorem states that no three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer value of n greater than two....
by factorisation using the roots of unity was for a very good reason: a failure of the fundamental theorem of arithmetic
Fundamental theorem of arithmetic
In number theory, the fundamental theorem of arithmetic states that any integer greater than 1 can be written as a unique product of prime numbers...
to hold in the ring
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
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 factorisation. We now recognise this as part of the ideal class group: in fact Kummer had isolated the p-torsion
Torsion subgroup
In the theory of abelian groups, the torsion subgroup AT of an abelian group A is the subgroup of A consisting of all elements that have finite order...
in that group for the 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 Dedekind
Richard Dedekind
Julius Wilhelm Richard Dedekind was a German mathematician who did important work in abstract algebra , algebraic number theory and the foundations of the real numbers.-Life:...
formulated the concept of ideal
Ideal (ring theory)
In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. The ideal concept allows the generalization in an appropriate way of some important properties of integers like "even number" or "multiple of 3"....
, 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
Algebraic integer
In number theory, an algebraic integer is a complex number that is a root of some monic polynomial with coefficients in . The set of all algebraic integers is closed under addition and multiplication and therefore is a subring of complex numbers denoted by A...
s do not always have unique factorization into primes (because they need not be principal ideal domain
Principal ideal domain
In abstract algebra, a principal ideal domain, or PID, is an integral domain in which every ideal is principal, i.e., can be generated by a single element. More generally, a principal ideal ring is a nonzero commutative ring whose ideals are principal, although some authors refer to PIDs as...
s), they do have the property that every proper ideal admits a unique factorization as a product of prime ideal
Prime ideal
In algebra , a prime ideal is a subset of a ring which shares many important properties of a prime number in the ring of integers...
s (that is, every ring of algebraic integers is a Dedekind domain
Dedekind domain
In abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors into a product of prime ideals. It can be shown that such a factorization is then necessarily unique up to the order of the factors...
). The size of the ideal class group can be considered as a measure for the deviation of a ring from being a principal domain; a ring is a principal domain if and only if it has a trivial ideal class group.
Definition
If R is an integral domain, define a relationRelation (mathematics)
In set theory and logic, a relation is a property that assigns truth values to k-tuples of individuals. Typically, the property describes a possible connection between the components of a k-tuple...
~ on nonzero fractional ideal
Fractional ideal
In mathematics, in particular commutative algebra, the concept of fractional ideal is introduced in the context of integral domains and is particularly fruitful in the study of Dedekind domains. In some sense, fractional ideals of an integral domain are like ideals where denominators are allowed...
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
Principal ideal
In ring theory, a branch of abstract algebra, a principal ideal is an ideal I in a ring R that is generated by a single element a of R.More specifically:...
of R consisting of all the multiples of a.) It is easily shown that this is an equivalence relation
Equivalence relation
In mathematics, an equivalence relation is a relation that, loosely speaking, partitions a set so that every element of the set is a member of one and only one cell of the partition. Two elements of the set are considered equivalent if and only if they are elements of the same cell...
. 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
Identity element
In mathematics, an identity element is a special type of element of a set with respect to a binary operation on that set. It leaves other elements unchanged when combined with them...
for this multiplication. Thus a class [I] has an 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
Monoid
In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single associative binary operation and an identity element. Monoids are studied in semigroup theory as they are naturally semigroups with identity. Monoids occur in several branches of mathematics; for...
.
However, if R is the ring of algebraic integers in an algebraic number field
Algebraic number field
In mathematics, an algebraic number field F is a finite field extension of the field of rational numbers Q...
, or more generally a Dedekind domain
Dedekind domain
In abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors into a product of prime ideals. It can be shown that such a factorization is then necessarily unique up to the order of the factors...
, the multiplication defined above turns the set of fractional ideal classes into an abelian group
Abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...
, the ideal class group of R. The group property of existence of inverse element
Inverse element
In abstract algebra, the idea of an inverse element generalises the concept of a negation, in relation to addition, and a reciprocal, in relation to multiplication. The intuition is of an element that can 'undo' the effect of combination with another given element...
s follows easily from the fact that, in a Dedekind domain, every non-zero ideal (except R) is a product of prime ideals.
Properties
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 domainPrincipal ideal domain
In abstract algebra, a principal ideal domain, or PID, is an integral domain in which every ideal is principal, i.e., can be generated by a single element. More generally, a principal ideal ring is a nonzero commutative ring whose ideals are principal, although some authors refer to PIDs as...
, and hence from satisfying unique prime factorization (Dedekind domains are unique factorization domain
Unique factorization domain
In mathematics, a unique factorization domain is, roughly speaking, a commutative ring in which every element, with special exceptions, can be uniquely written as a product of prime elements , analogous to the fundamental theorem of arithmetic for the integers...
s if and only if they are principal ideal domains).
The number of ideal classes (the class number of R) may be infinite in general. In fact, every abelian group is isomorphic to the ideal class group of some Dedekind domain. But if R is in fact 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
Algebraic number field
In mathematics, an algebraic number field F is a finite field extension of the field of rational numbers Q...
of small discriminant
Discriminant of an algebraic number field
In mathematics, the discriminant of an algebraic number field is a numerical invariant that, loosely speaking, measures the size of the algebraic number field...
, using Minkowski's bound
Minkowski's bound
In algebraic number theory, Minkowski's bound gives an upper bound of the norm of ideals to be checked in order to determine the class number of a number field K...
. This result gives a bound, depending on the ring, such that every ideal class contains an ideal of norm
Norm of an ideal
In commutative algebra, the norm of an ideal is a generalization of a norm of an element in the field extension. It is particularly important in number theory since it measures the size of an ideal of a complicated number ring in terms of an ideal in a less complicated ring...
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 functorial, and the class group can be subsumed under the heading of algebraic K-theory
Algebraic K-theory
In mathematics, algebraic K-theory is an important part of homological algebra concerned with defining and applying a sequenceof functors from rings to abelian groups, for all integers n....
, with K0(R) being the functor assigning to R its ideal class group; more precisely, K0(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 domainDedekind domain
In abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors into a product of prime ideals. It can be shown that such a factorization is then necessarily unique up to the order of the factors...
behave like elements. The other part of the answer is provided by the multiplicative group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
of unit
Unit (ring theory)
In mathematics, an invertible element or a unit in a ring R refers to any element u that has an inverse element in the multiplicative monoid of R, i.e. such element v that...
s of the Dedekind domain, since passage from principal ideals
to their generators requires the use of units (and this is the rest of the reason for introducing the concept of fractional ideal, as well):
Define a map from K× 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
Group homomorphism
In mathematics, given two groups and , a group homomorphism from to is a function h : G → H such that for all u and v in G it holds that h = h \cdot h...
; its kernel
Kernel (algebra)
In the various branches of mathematics that fall under the heading of abstract algebra, the kernel of a homomorphism measures the degree to which the homomorphism fails to be injective. An important special case is the kernel of a matrix, also called the null space.The definition of kernel takes...
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 Z, Z[ω], and Z[i], where ω is a cube root of 1 and i is a fourth root of 1 (i.e. a square root of −1), are all principal ideal domains, and so have class number 1: that is, they have trivial ideal class groups.
- If k is a field, then the polynomial ringPolynomial ringIn mathematics, especially in the field of abstract algebra, a polynomial ring is a ring formed from the set of polynomials in one or more variables with coefficients in another ring. Polynomial rings have influenced much of mathematics, from the Hilbert basis theorem, to the construction of...
k[X1, X2, X3, ...] is an integral domain. It has a countably infinite set of ideal classes.
Class numbers of quadratic fields
If d is a square-free integerSquare-free integer
In mathematics, a square-free, or quadratfrei, integer is one divisible by no perfect square, except 1. For example, 10 is square-free but 18 is not, as it is divisible by 9 = 32...
(a product of distinct primes) other than 1, then Q(√d) is a quadratic extension of Q
Quadratic field
In algebraic number theory, a quadratic field is an algebraic number field K of degree two over Q. It is easy to show that the map d ↦ Q is a bijection from the set of all square-free integers d ≠ 0, 1 to the set of all quadratic fields...
. If d < 0, then the class number of the ring R of algebraic integers of Q(√d) is equal to 1 for precisely the following values of d: d = −1, −2, −3, −7, −11, −19, −43, −67, and −163. This result was first conjectured by Gauss
Carl Friedrich Gauss
Johann Carl Friedrich Gauss was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy and optics.Sometimes referred to as the Princeps mathematicorum...
and proven by Kurt Heegner
Kurt Heegner
Kurt Heegner was a German private scholar from Berlin, who specialized inradio engineering and mathematics...
, although Heegner's proof was not believed until Harold Stark
Harold Stark
Harold Mead Stark is an American mathematician, specializing in number theory. He is best known for his solution of the Gauss class number 1 problem, in effect correcting and completing the earlier work of Kurt Heegner; and for Stark's conjecture. He has recently collaborated with Audrey Terras on...
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 Q(√d) 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 fields with class number 1.
For d < 0, the ideal class group of Q(√d) is isomorphic to the class group of integral binary quadratic form
Binary quadratic form
In mathematics, a binary quadratic form is a quadratic form in two variables. More concretely, it is a homogeneous polynomial of degree 2 in two variableswhere a, b, c are the coefficients...
s of discriminant
Discriminant
In algebra, the discriminant of a polynomial is an expression which gives information about the nature of the polynomial's roots. For example, the discriminant of the quadratic polynomialax^2+bx+c\,is\Delta = \,b^2-4ac....
equal to the discriminant of Q(√d). 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
Narrow class group
In algebraic number theory, the narrow class group of a number field K is a refinement of the class group of K that takes into account some information about embeddings of K into the field of real numbers.- Formal definition :...
of Q(√d).
Example of a non-trivial class group
The quadratic integerQuadratic integer
In number theory, quadratic integers are a generalization of the rational integers to quadratic fields. Important examples include the Gaussian integers and the Eisenstein integers. Though they have been studied for more than a hundred years, many open problems remain.- Definition :Quadratic...
ring R = Z [√−5] is the ring of integers of Q(√−5). It does not possess unique factorization; in fact the class group of R is cyclic of order 2. Indeed, the ideal
- J = (2, 1 + √−5)
is not principal, which can be proved by contradiction as follows. If J were generated by an element x of R, then x would divide both 2 and 1 + √−5. Then the norm
Field norm
In mathematics, the norm is a mapping defined in field theory, to map elements of a larger field into a smaller one.-Formal definitions:1. Let K be a field and L a finite extension of K...
N(x) of x would divide both N(2) = 4 and N(1 + √−5) = 6, so N(x) would divide 2. We are assuming that x is not a unit of R, so N(x) cannot be 1. It cannot be 2 either, because R has no elements of norm 2, that is, the equation b2 + 5c2 = 2 has no solutions in integers.
One also computes that J2 = (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.
The fact that this J is not principal is also related to the fact that the element 6 has two distinct factorisations into irreducibles:
- 6 = 2 × 3 = (1 + √−5) × (1 − √−5).
Connections to class field theory
Class field theoryClass field theory
In mathematics, class field theory is a major branch of algebraic number theory that studies abelian extensions of number fields.Most of the central results in this area were proved in the period between 1900 and 1950...
is a branch of algebraic number theory
Algebraic number field
In mathematics, an algebraic number field F is a finite field extension of the field of rational numbers Q...
which seeks to classify all the abelian extension
Galois theory
In mathematics, more specifically in abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory and group theory...
s of a given algebraic number field, meaning Galois extensions with abelian Galois group
Galois group
In mathematics, more specifically in the area of modern algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension...
. A particularly beautiful example is found in the Hilbert class field
Hilbert class field
In algebraic number theory, the Hilbert class field E of a number field K is the maximal abelian unramified extension of K. Its degree over K equals the class number of K and the Galois group of E over K is canonically isomorphic to the ideal class group of K using Frobenius elements for prime...
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 formulaClass number formulaIn number theory, the class number formula relates many important invariants of a number field to a special value of its Dedekind zeta function-General statement of the class number formula:...
- Class number problem
- List of number fields with class number one
- Principal ideal domainPrincipal ideal domainIn abstract algebra, a principal ideal domain, or PID, is an integral domain in which every ideal is principal, i.e., can be generated by a single element. More generally, a principal ideal ring is a nonzero commutative ring whose ideals are principal, although some authors refer to PIDs as...
- Algebraic K-theoryAlgebraic K-theoryIn mathematics, algebraic K-theory is an important part of homological algebra concerned with defining and applying a sequenceof functors from rings to abelian groups, for all integers n....
- Galois theoryGalois theoryIn mathematics, more specifically in abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory and group theory...
- Fermat's last theoremFermat's Last TheoremIn number theory, Fermat's Last Theorem states that no three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer value of n greater than two....
- Narrow class groupNarrow class groupIn algebraic number theory, the narrow class group of a number field K is a refinement of the class group of K that takes into account some information about embeddings of K into the field of real numbers.- Formal definition :...
- Picard group—a generalisation of the class group appearing in algebraic geometryAlgebraic geometryAlgebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...