Algebraic number theory

Algebraic number theory

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 [√Overline|-5] where: 6=2cdot3=(1+sqrt{-5})cdot(1-sqrt{-5}).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"):="p"Z 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:2mathbf{Z} [i] =left((1+i)mathbf{Z} [i] ight)^2.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":pmathbf{Z} [i] mbox{ is a prime ideal if }pequiv 3 ,(operatorname{mod}, 4):pmathbf{Z} [i] mbox{ is not a prime ideal if }pequiv 1 ,(operatorname{mod}, 4).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 |·| : QR, there are absolute value functions |·|p : QR 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: mathfrak{p}-adic absolute values like |·|p, one for each prime ideal mathfrak{p} 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 |·| : CR. 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 { 2, 3, 5, 7, ..., ∞ }, and the usual absolute value on Q is often denoted |·| 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 Overline|τ, called the "conjugate embedding", by composing τ with the complex conjugation map CC. Given such a pair of embeddings τ and Overline|τ, there corresponds a unique prime of "K" again obtained by composing τ with the usual absolute value (composing Overline|τ instead gives the same absolute value function since |"z"| = |Overline|"z"| for any complex number "z", where Overline|"z" 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, Overline|τ : "K" → C. The number of real (respectively, complex) primes is often denoted "r"1 (respectively, "r"2). Then, the total number of embeddings "K" → C is "r"1+2"r"2 (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 its structure as a Galois module for the Galois group of "K"/Q is also possible. [See proposition VIII.8.6.11 of Neukirch et al. 2000] 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.

Major results

Finiteness of the class group

One of the class 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" × Z"r", where "G" is the finite cyclic group consisting of all the roots of unity in "O", and "r" = "r"1 + "r"2 − 1 (where "r"1 (respectively, "r"2) 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 "r"1 + "r"2 − 1 whose torsion consists of the roots of unity in "O".

Artin reciprocity

Class number formula

Notes

References

Introductory texts

* Kenneth Ireland and Michael Rosen, "A Classical Introduction to Modern Number Theory, Second Edition", Springer-Verlag, 1990
* Ian Stewart and David Tall, "Algebraic Number Theory and Fermat's Last Theorem," A. K. Peters, 2002

Intermediate texts

* Daniel A. Marcus, "Number Fields"

Graduate level accounts

*Citation
editor-last=Cassels
editor-first=J. W. S.
editor-link=J. W. S. Cassels
editor2-last=Frölich
editor2-first=Albrecht
editor2-link=Albrecht Fröhlich
title=Algebraic number theory
year=1967
place=London
publisher=Academic Press
id=MathSciNet | id = 0215665

*Citation
last=Fröhlich
first=Albrecht
author-link=Albrecht Fröhlich
last2=Taylor
first2=Martin J.
author2-link=Martin J. Taylor
title=Algebraic number theory
publisher=Cambridge University Press
year=1993
series=Cambridge Studies in Advanced Mathematics
volume=27
isbn=0-521-43834-9
id=MathSciNet | id = 1215934

*Citation
last=Lang
first=Serge
author-link=Serge Lang
title=Algebraic number theory
edition=2
publisher=Springer-Verlag
year=1994
series=Graduate Texts in Mathematics
volume=110
place=New York
isbn=0-387-94225-4
id=MathSciNet | id = 1282723

*Neukirch ANT

pecific references

* | year=2000 | volume=323

ee also

*Arithmétique modulaire A survey of number theory, with applications (in French Wikipedia)


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