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 abelian group known as an ideal class group (or class group). Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, a unique factorization domain (UFD is roughly speaking a Commutative ring in which every element with special exceptions can be uniquely written In Mathematics, an algebraic number field (or simply number field) F is a finite (and hence algebraic) Field extension of the In Abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an Integral domain in which every nonzero Proper An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the 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. In Group theory, a branch of Mathematics, the term order is used in two closely related senses the order of a group is 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. In Mathematics, a unique factorization domain (UFD is roughly speaking a Commutative ring in which every element with special exceptions can be uniquely written
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History and origin of the ideal class group
The first ideal class groups encountered in mathematics were part of the theory of quadratic forms: 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. In Mathematics, a quadratic form is a Homogeneous polynomial of degree two in a number of variables Johann Carl Friedrich Gauss (ˈɡaʊs, Gauß Carolus Fridericus Gauss ( 30 April 1777 – 23 February 1855) was a German This gave a finite abelian group, as was recognised at the time. An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the
Later Kummer was working towards a theory of cyclotomic fields. Ernst Eduard Kummer ( 29 January 1810 - 14 May 1893) was a German Mathematician. In Number theory, a cyclotomic field is a Number field obtained by adjoining a complex Root of unity to Q, the field of Rational numbers 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 rings generated by those roots of unity, was a major obstacle. Fermat's Last Theorem is the name of the statement in Number theory that It is impossible to separate any power higher than the second into two like In Mathematics, the n th roots of unity, or de Moivre numbers are all the Complex numbers that yield 1 when raised to a given power In Number theory, the fundamental theorem of arithmetic (or unique-prime-factorization theorem) states that every Natural number greater than 1 can be written In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real 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). In the theory of Abelian groups the torsion subgroup AT of an abelian group A is the Subgroup of A consisting of all elements In Number theory, a regular prime is a certain kind of Prime number.
Somewhat later again Dedekind formulated the concept of ideal, Kummer having worked in a different way. Julius Wilhelm Richard Dedekind ( October 6, 1831 &ndash February 12, 1916) was a German mathematician who did important In Ring theory, a branch of Abstract algebra, an ideal is a special Subset of a ring. At this point the existing examples could be unified. It was shown that while rings of algebraic integers do not always have unique factorization into primes (because they need not be principal ideal domains), they do have the property that every proper ideal admits a unique factorization as a product of prime ideals (that is, every ring of algebraic integers is a Dedekind domain). This article deals with the ring of complex numbers integral over Z. In Number theory, the fundamental theorem of arithmetic (or unique-prime-factorization theorem) states that every Natural number greater than 1 can be written In Abstract algebra, a principal ideal domain, or PID is an Integral domain in which every ideal is principal i In Mathematics, a prime ideal is a Subset of a ring which shares many important properties of a Prime number in the Ring of integers In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real In Abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an Integral domain in which every nonzero Proper The ideal class group gives some answer to the question: which ideals are principal ideals? The answer comes in the form all of them, if and only if the ideal class group (which is a finite group) has just one element. 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
Technical development
If R is an integral domain, define a relation ~ on nonzero fractional(!) ideals of R by I ~ J whenever there exist nonzero elements a and b of R such that (a)I = (b)J. In Abstract algebra, a branch of Mathematics, an integral domain is a Commutative ring with an additive identity 0 and a multiplicative identity 1 such This article sets out the set-theoretic notion of relation For a more elementary point of view see Binary relations and Triadic relations (Here the notation (a) means the principal ideal of R consisting of all the multiples of a. 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 ) It is easily shown that this is an equivalence relation. In Mathematics, an equivalence relation is a Binary relation between two elements of a set which groups them together as being "equivalent" The equivalence classes are called the ideal classes of R. In Mathematics, given a set X and an Equivalence relation ~ on X, the equivalence class of an element a in X 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. In Mathematics, commutativity is the ability to change the order of something without changing the end result The principal ideals form the ideal class [R] which serves as an identity element for this multiplication. In Mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a Binary operation on that 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. In Abstract algebra, a branch of Mathematics, a monoid is an Algebraic structure with a single Associative Binary operation
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. This article deals with the ring of complex numbers integral over Z. In Mathematics, an algebraic number field (or simply number field) F is a finite (and hence algebraic) Field extension of the In Abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an Integral domain in which every nonzero Proper An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the 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 ideals. In Mathematics, the idea of inverse element generalises the concepts of negation, in relation to Addition, and reciprocal, in relation to In Mathematics, a prime ideal is a Subset of a ring which shares many important properties of a Prime number in the Ring of integers
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 domains if and only if they are principal ideal domains). In Abstract algebra, a principal ideal domain, or PID is an Integral domain in which every ideal is principal i In Mathematics, a unique factorization domain (UFD is roughly speaking a Commutative ring in which every element with special exceptions can be uniquely written
The number of ideal classes (the class number of R) may be infinite in general. But if R is in fact a ring of algebraic integers, then this 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 a theorem of Minkowski. In Mathematics, an algebraic number field (or simply number field) F is a finite (and hence algebraic) Field extension of the In Mathematics, the discriminant of an Algebraic number field is a numerical invariant that loosely speaking measures the size of the ( Ring of integers Hermann Minkowski ( June 22 1864 – January 12 1909) was a Russian born German Mathematician, of Jewish This result gives a bound, depending on the ring, such that every ideal class contains an ideal of norm less than the bound. The norm of an ideal is defined in Algebraic number theory. Let K\subset L be two number fields with rings of integers O_K\subset O_L 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.
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. In Abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an Integral domain in which every nonzero Proper The other part of the answer is provided by the multiplicative group of units 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). In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Mathematics, a unit in a ( Unital) ring R is an invertible element of R, i
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. In Mathematics, given two groups ( G, * and ( H, · a group homomorphism from ( G, * to ( H, · is a function 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 The failure of these groups to be trivial is a precise 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.
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. In Mathematics, algebraic K-theory is an advanced part of Homological algebra concerned with defining and applying a sequence K n Higher K groups can also be employed and interpreted arithmetically in connection to rings of integers.
Examples of ideal class groups
The rings Z, Z[i], and Z[ω], (where i is a square root of -1 and ω is a cube 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 ring k[X1, X2, X3, . In 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 . . ] is an integral domain. It has a countably infinite set of ideal classes.
If d is a square-free integer (in other words, a product of distinct primes) other than 1, then Q(√d) is a finite extension of Q. In Mathematics, an element r of a Unique factorization domain R is called square-free if it is not divisible by a non-trivial square In particular it is a 2-dimensional vector space over Q, called a quadratic field. In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added In Mathematics, a quadratic field is an Algebraic number field K of degree two over 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, which Stark showed was actually equivalent to Heegner's. Johann Carl Friedrich Gauss (ˈɡaʊs, Gauß Carolus Fridericus Gauss ( 30 April 1777 – 23 February 1855) was a German Kurt Heegner (1893–1965 was a German private scholar from Berlin, who specialized in Radio Engineering and Mathematics. Harold Mead Stark (born 1939) is an American Mathematician, specializing in Number theory. (See Stark-Heegner theorem. In Number theory, a branch of Mathematics, the Stark–Heegner theorem states precisely which quadratic imaginary number fields admit unique factorisation ) This is a special case of the famous class number problem. In Mathematics, the Gauss class number problem ( for imaginary quadratic fields) as is usually understood is to provide for each n &ge 1 a complete
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 to say the least.
Example of a non-trivial class group
The ring R = Z [√−5] is the ring of integers of Q(√−5). It does not possess unique factorisation; 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 N(x) of x would divide both N(2) = 4 and N(1 + √−5) = 6, so N(x) would divide 2. In Mathematics, the (field norm is a mapping defined in field theory, to map elements of a larger field into a smaller one 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 theory is a branch of algebraic number theory which seeks to classify all the abelian extensions of a given algebraic number field, meaning Galois extensions with abelian Galois group. In Mathematics, class field theory is a major branch of Algebraic number theory. In Mathematics, an algebraic number field (or simply number field) F is a finite (and hence algebraic) Field extension of the In Mathematics, more specifically in Abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory In Mathematics, an algebraic number field (or simply number field) F is a finite (and hence algebraic) Field extension of the In Mathematics, a Galois group is a group associated with a certain type of Field extension. 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. In Algebraic number theory, the Hilbert class field E of a Number field K is the Maximal abelian Unramified In Mathematics, ramification is a geometric term used for 'branching out' in the way that the Square root function for Complex numbers can be seen 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
- Ring
- Ideal
- 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
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