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Algebraic number theory
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In mathematics, algebraic number theory is a major branch of number theory which studies the algebraic structures related to algebraic integers. This is generally accomplished by considering a ring of algebraic integers O in an algebraic number field K/Q (i.e. a finite extension of the rational numbers Q), and studying the properties of these rings and fields (e.g. factorization, ideals, field extensions). In this setting, the familiar features of the integers (e.g. unique factorization) need not hold.
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Encyclopedia
In mathematics, algebraic number theory is a major branch of number theory which studies the algebraic structures related to algebraic integers. This is generally accomplished by considering a ring of algebraic integers O in an algebraic number field K/Q (i.e. a finite extension of the rational numbers Q), and studying the properties of these rings and fields (e.g. factorization, ideals, field extensions). In this setting, the familiar features of the integers (e.g. unique factorization) need not hold. The virtue of the machinery employed — Galois theory, group cohomology, group representations and L-functions — is that it allows one to recover that order partly for this new class of integers.
Basic notions
Unique factorization and the ideal class group
One of the first properties of Z that can fail in the ring of integers O of an algebraic number field K is that of the unique factorization of integers into prime numbers. The prime numbers in Z are generalized to irreducible elements in O, and though the unique factorization of elements of O into irreducible elements may hold in some cases (such as for the Gaussian integers Z[i]), it may also fail, as in the case of Z[√] where
The ideal class group of O is a measure of how much unique factorization of elements fails; in particular, the ideal class group is trivial if, and only if, O is a unique factorization domain.
Factoring prime ideals in extensions
Unique factorization can be partially recovered for O in that it has the property of unique factorization of ideals into prime ideals (i.e. it is a Dedekind domain). This makes the study of the prime ideals in O particularly important. This is another area where things change from Z to O: the prime numbers, which generate prime ideals of Z (in fact, every single prime ideal of Z is of the form (p):=pZ for some prime number p,) may no longer generate prime ideals in O. For example, in the ring of Gaussian integers, the ideal 2Z[i] is no longer a prime ideal; in fact
On the other hand, the ideal 3Z[i] is a prime ideal. The complete answer for the Gaussian integers is obtained by using a theorem of Fermat's, with the result being that for an odd prime number p
Generalizing this simple result to more general rings of integers is a basic problem in algebraic number theory. Class field theory accomplishes this goal when K is an abelian extension of Q (i.e. a Galois extension with abelian Galois group).
Primes and places
An important generalization of the notion of prime ideal in O is obtained by passing from the so-called ideal-theoretic approach to the so-called valuation-theoretic approach. The relation between the two approaches arises as follows. In addition to the usual absolute value function |·| : Q ? R, there are absolute value functions |·|p : Q ? R defined for each prime number p in Z, called p-adic absolute values. Ostrowski's theorem states that these are all possible absolute value functions on Q (up to equivalence). This suggests that the usual absolute value could be considered as another prime. More generally, a prime of an algebraic number field K (also called a place) is an equivalence class of absolute values on K. The primes in K are of two sorts: -adic absolute values like |·|p, one for each prime ideal of O, and absolute values like |·| obtained by considering K as a subset of the complex numbers in various possible ways and using the absolute value |·| : C ? R. A prime of the first kind is called a finite prime (or finite place) and one of the second kind is called an infinite prime (or infinite place). Thus, the set of primes of Q is generally denoted , and the usual absolute value on Q is often denoted |·|8 in this context.
The set of infinite primes of K can be described explicitly in terms of the embeddings K ? C (i.e. the non-zero ring homomorphisms from K to C). Specifically, the set of embeddings can be split up into two disjoint subsets, those whose image is contained in R, and the rest. To each embedding σ : K ? R, there corresponds a unique prime of K coming from the absolute value obtained by composing σ with the usual absolute value on R; a prime arising in this fashion is called a real prime (or real place). To an embedding τ : K ? C whose image is not contained in R, one can construct a distinct embedding , called the conjugate embedding, by composing τ with the complex conjugation map C ? C. Given such a pair of embeddings τ and , there corresponds a unique prime of K again obtained by composing τ with the usual absolute value (composing instead gives the same absolute value function since |z| = || for any complex number z, where denotes the complex conjugate of z). Such a prime is called a complex prime (or complex place). The description of the set of infinite primes is then as follows: each infinite prime corresponds either to a unique embedding σ : K ? R, or a pair of conjugate embeddings τ, : K ? C. The number of real (respectively, complex) primes is often denoted r1 (respectively, r2). Then, the total number of embeddings K ? C is r1+2r2 (which, in fact, equals the degree of the extension K/Q).
Units
The fundamental theorem of arithmetic describes the multiplicative structure of Z. It states that every non-zero integer can be written (essentially) uniquely as a product of prime powers and ±1. The unique factorization of ideals in the ring O recovers part of this description, but fails to address the factor ±1. The integers 1 and -1 are the invertible elements (i.e. units) of Z. More generally, the invertible elements in O form a group under multiplication called the unit group of O, denoted O×. This group can be much larger than the cyclic group of order 2 formed by the units of Z. Dirichlet's unit theorem describes the abstract structure of the unit group as an abelian group. A more precise statement giving the structure of O× ⊗Z Q as a Galois module for the Galois group of K/Q is also possible. The size of the unit group, and its lattice structure give important numerical information about O, as can be seen in the class number formula.
Local fields
Completing a number field at a place w gives a complete field. If the valuation is archimedean, one gets R or C, if it is non-archimedean and lies over a prime p of the rationals, one gets a finite extension Kw / Qp: a complete, discrete valued field with finite residue field. This process simplifies the arithmetic of the field and allows the local study of problems. For example the Kronecker-Weber theorem can be deduced easily from the analogous local statement.
Major results
Finiteness of the class group
One of the classical results in algebraic number theory is that the ideal class group of an algebraic number field K is finite. The order of the class group is called the class number, and is often denoted by the letter h.
Dirichlet's unit theorem
Dirichlet's unit theorem provides a description of the structure of the multiplicative group of units O× of the ring of integers O. Specifically, it states that O× is isomorphic to G × Zr, where G is the finite cyclic group consisting of all the roots of unity in O, and r = r1 + r2 - 1 (where r1 (respectively, r2) denotes the set of real embeddings (respectively, pairs of conjugate non-real embeddings) of K). In other words, O× is a finitely generated abelian group of rank r1 + r2 - 1 whose torsion consists of the roots of unity in O.
Artin reciprocity
See also
- Quadratic reciprocity
- Cubic reciprocity
- Quartic reciprocity
Class number formula
Introductory texts
- Kenneth Ireland and Michael Rosen, "A Classical Introduction to Modern Number Theory, Second Edition", Springer-Verlag, 1990
- Ian Stewart and David O. Tall, "Algebraic Number Theory and Fermat's Last Theorem," A. K. Peters, 2002
Intermediate texts
- Daniel A. Marcus, "Number Fields"
Graduate level accounts
Specific references
See also
- Arithmétique modulaire A survey of number theory, with applications (in French Wikipedia)
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