- Algebraic number theory
In

mathematics ,**algebraic number theory**is a major branch ofnumber theory which studies thealgebraic structure s related toalgebraic integer s. This is generally accomplished by considering a ring of algebraic integers "O" in analgebraic number field "K"/**Q**(i.e. a finite extension of therational number s**Q**), and studying the properties of these rings and fields (e.g.factorization , ideals, field extensions). In this setting, the familiar features of theinteger s (e.g.unique factorization ) need not hold. The virtue of the machinery employed —Galois theory ,group cohomology ,group representation s andL-function s — 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 thering of integers "O" of an algebraic number field "K" is that of the unique factorization of integers intoprime number s. The prime numbers in**Z**are generalized toirreducible element s in "O", and though the unique factorization of elements of "O" into irreducible elements may hold in some cases (such as for theGaussian integers **Z**[i] ), it may also fail, as in the case of**Z**[√Overline|-5] where:$6=2cdot3=(1+sqrt\{-5\})cdot(1-sqrt\{-5\}).$Theideal 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 aunique 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 ideal s (i.e. it is aDedekind 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 2**Z**[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 3**Z**[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 anabelian extension of**Q**(i.e. aGalois extension with abelianGalois 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**, calledp-adic absolute value s.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 anequivalence 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 thecomplex number s 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 { 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-zeroring homomorphism s 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 thecomplex conjugation map**C**→**C**. 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 ofprime power s 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 thecyclic 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 aGalois 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 theclass 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 afinitely 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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