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**Algebraic number theory** is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic objects such as algebraic number fields and their rings of integers, finite fields, and function fields. These properties, such as whether a ring admits unique factorization, the behavior of ideals, and the Galois groups of fields, can resolve questions of primary importance in number theory, like the existence of solutions to Diophantine equations.

- History of algebraic number theory
- Diophantus
- Fermat
- Gauss
- Dirichlet
- Dedekind
- Hilbert
- Artin
- Modern theory
- Basic notions
- Failure of unique factorization
- Factorization into prime ideals
- Ideal class group
- Real and complex embeddings
- Places
- Units
- Zeta function
- Local fields
- Major results
- Finiteness of the class group
- Dirichlet's unit theorem
- Reciprocity laws
- Class number formula
- Related areas
- See also
- Notes
- Further reading
- Introductory texts
- Intermediate texts
- Graduate level texts
- External links

The beginnings of algebraic number theory can be traced to Diophantine equations,^{ [1] } named after the 3rd-century Alexandrian mathematician, Diophantus, who studied them and developed methods for the solution of some kinds of Diophantine equations. A typical Diophantine problem is to find two integers *x* and *y* such that their sum, and the sum of their squares, equal two given numbers *A* and *B*, respectively:

Diophantine equations have been studied for thousands of years. For example, the solutions to the quadratic Diophantine equation *x*^{2} + *y*^{2} = *z*^{2} are given by the Pythagorean triples, originally solved by the Babylonians (c. 1800 BC).^{ [2] } Solutions to linear Diophantine equations, such as 26*x* + 65*y* = 13, may be found using the Euclidean algorithm (c. 5th century BC).^{ [3] }

Diophantus' major work was the * Arithmetica *, of which only a portion has survived.

Fermat's last theorem was first conjectured by Pierre de Fermat in 1637, famously in the margin of a copy of *Arithmetica* where he claimed he had a proof that was too large to fit in the margin. No successful proof was published until 1995 despite the efforts of countless mathematicians during the 358 intervening years. The unsolved problem stimulated the development of algebraic number theory in the 19th century and the proof of the modularity theorem in the 20th century.

One of the founding works of algebraic number theory, the * Disquisitiones Arithmeticae* (Latin:

The *Disquisitiones* was the starting point for the work of other nineteenth century European mathematicians including Ernst Kummer, Peter Gustav Lejeune Dirichlet and Richard Dedekind. Many of the annotations given by Gauss are in effect announcements of further research of his own, some of which remained unpublished. They must have appeared particularly cryptic to his contemporaries; we can now read them as containing the germs of the theories of L-functions and complex multiplication, in particular.

In a couple of papers in 1838 and 1839 Peter Gustav Lejeune Dirichlet proved the first class number formula, for quadratic forms (later refined by his student Leopold Kronecker). The formula, which Jacobi called a result "touching the utmost of human acumen", opened the way for similar results regarding more general number fields.^{ [5] } Based on his research of the structure of the unit group of quadratic fields, he proved the Dirichlet unit theorem, a fundamental result in algebraic number theory.^{ [6] }

He first used the pigeonhole principle, a basic counting argument, in the proof of a theorem in diophantine approximation, later named after him Dirichlet's approximation theorem. He published important contributions to Fermat's last theorem, for which he proved the cases *n* = 5 and *n* = 14, and to the biquadratic reciprocity law.^{ [5] } The Dirichlet divisor problem, for which he found the first results, is still an unsolved problem in number theory despite later contributions by other researchers.

Richard Dedekind's study of Lejeune Dirichlet's work was what led him to his later study of algebraic number fields and ideals. In 1863, he published Lejeune Dirichlet's lectures on number theory as * Vorlesungen über Zahlentheorie * ("Lectures on Number Theory") about which it has been written that:

"Although the book is assuredly based on Dirichlet's lectures, and although Dedekind himself referred to the book throughout his life as Dirichlet's, the book itself was entirely written by Dedekind, for the most part after Dirichlet's death." (Edwards 1983)

1879 and 1894 editions of the *Vorlesungen* included supplements introducing the notion of an ideal, fundamental to ring theory. (The word "Ring", introduced later by Hilbert, does not appear in Dedekind's work.) Dedekind defined an ideal as a subset of a set of numbers, composed of algebraic integers that satisfy polynomial equations with integer coefficients. The concept underwent further development in the hands of Hilbert and, especially, of Emmy Noether. Ideals generalize Ernst Eduard Kummer's ideal numbers, devised as part of Kummer's 1843 attempt to prove Fermat's Last Theorem.

David Hilbert unified the field of algebraic number theory with his 1897 treatise * Zahlbericht * (literally "report on numbers"). He also resolved a significant number-theory problem formulated by Waring in 1770. As with the finiteness theorem, he used an existence proof that shows there must be solutions for the problem rather than providing a mechanism to produce the answers.^{ [7] } He then had little more to publish on the subject; but the emergence of Hilbert modular forms in the dissertation of a student means his name is further attached to a major area.

He made a series of conjectures on class field theory. The concepts were highly influential, and his own contribution lives on in the names of the Hilbert class field and of the Hilbert symbol of local class field theory. Results were mostly proved by 1930, after work by Teiji Takagi.^{ [8] }

Emil Artin established the Artin reciprocity law in a series of papers (1924; 1927; 1930). This law is a general theorem in number theory that forms a central part of global class field theory.^{ [9] } The term "reciprocity law" refers to a long line of more concrete number theoretic statements which it generalized, from the quadratic reciprocity law and the reciprocity laws of Eisenstein and Kummer to Hilbert's product formula for the norm symbol. Artin's result provided a partial solution to Hilbert's ninth problem.

Around 1955, Japanese mathematicians Goro Shimura and Yutaka Taniyama observed a possible link between two apparently completely distinct, branches of mathematics, elliptic curves and modular forms. The resulting modularity theorem (at the time known as the Taniyama–Shimura conjecture) states that every elliptic curve is modular, meaning that it can be associated with a unique modular form.

It was initially dismissed as unlikely or highly speculative, and was taken more seriously when number theorist André Weil found evidence supporting it, but no proof; as a result the "astounding"^{ [10] } conjecture was often known as the Taniyama–Shimura-Weil conjecture. It became a part of the Langlands program, a list of important conjectures needing proof or disproof.

From 1993 to 1994, Andrew Wiles provided a proof of the modularity theorem for semistable elliptic curves, which, together with Ribet's theorem, provided a proof for Fermat's Last Theorem. Almost every mathematician at the time had previously considered both Fermat's Last Theorem and the Modularity Theorem either impossible or virtually impossible to prove, even given the most cutting edge developments. Wiles first announced his proof in June 1993^{ [11] } in a version that was soon recognized as having a serious gap at a key point. The proof was corrected by Wiles, partly in collaboration with Richard Taylor, and the final, widely accepted version was released in September 1994, and formally published in 1995. The proof uses many techniques from algebraic geometry and number theory, and has many ramifications in these branches of mathematics. It also uses standard constructions of modern algebraic geometry, such as the category of schemes and Iwasawa theory, and other 20th-century techniques not available to Fermat.

An important property of the ring of integers is that it satisfies the fundamental theorem of arithmetic, that every (positive) integer has a factorization into a product of prime numbers, and this factorization is unique up to the ordering of the factors. This may no longer be true in the ring of integers *O* of an algebraic number field *K*.

A *prime element* is an element *p* of *O* such that if *p* divides a product *ab*, then it divides one of the factors *a* or *b*. This property is closely related to primality in the integers, because any positive integer satisfying this property is either 1 or a prime number. However, it is strictly weaker. For example, −2 is not a prime number because it is negative, but it is a prime element. If factorizations into prime elements are permitted, then, even in the integers, there are alternative factorizations such as

In general, if *u* is a unit, meaning a number with a multiplicative inverse in *O*, and if *p* is a prime element, then *up* is also a prime element. Numbers such as *p* and *up* are said to be *associate*. In the integers, the primes *p* and −*p* are associate, but only one of these is positive. Requiring that prime numbers be positive selects a unique element from among a set of associated prime elements. When *K* is not the rational numbers, however, there is no analog of positivity. For example, in the Gaussian integers **Z**[*i*],^{ [12] } the numbers 1 + 2*i* and −2 + *i* are associate because the latter is the product of the former by *i*, but there is no way to single out one as being more canonical than the other. This leads to equations such as

which prove that in **Z**[*i*], it is not true that factorizations are unique up to the order of the factors. For this reason, one adopts the definition of unique factorization used in unique factorization domains (UFDs). In a UFD, the prime elements occurring in a factorization are only expected to be unique up to units and their ordering.

However, even with this weaker definition, many rings of integers in algebraic number fields do not admit unique factorization. There is an algebraic obstruction called the ideal class group. When the ideal class group is trivial, the ring is a UFD. When it is not, there is a distinction between a prime element and an irreducible element. An *irreducible element**x* is an element such that if *x* = *yz*, then either *y* or *z* is a unit. These are the elements that cannot be factored any further. Every element in *O* admits a factorization into irreducible elements, but it may admit more than one. This is because, while all prime elements are irreducible, some irreducible elements may not be prime. For example, consider the ring **Z**[√-5].^{ [13] } In this ring, the numbers 3, 2 + √-5 and 2 - √-5 are irreducible. This means that the number 9 has two factorizations into irreducible elements,

This equation shows that 3 divides the product (2 + √-5)(2 - √-5) = 9. If 3 were a prime element, then it would divide 2 + √-5 or 2 - √-5, but it does not, because all elements divisible by 3 are of the form 3*a* + 3*b*√-5. Similarly, 2 + √-5 and 2 - √-5 divide the product 3^{2}, but neither of these elements divides 3 itself, so neither of them are prime. As there is no sense in which the elements 3, 2 + √-5 and 2 - √-5 can be made equivalent, unique factorization fails in **Z**[√-5]. Unlike the situation with units, where uniqueness could be repaired by weakening the definition, overcoming this failure requires a new perspective.

If *I* is an ideal in *O*, then there is always a factorization

where each is a prime ideal, and where this expression is unique up to the order of the factors. In particular, this is true if *I* is the principal ideal generated by a single element. This is the strongest sense in which the ring of integers of a general number field admits unique factorization. In the language of ring theory, it says that rings of integers are Dedekind domains.

When *O* is a UFD, every prime ideal is generated by a prime element. Otherwise, there are prime ideals which are not generated by prime elements. In **Z**[√-5], for instance, the ideal (2, 1 + √-5) is a prime ideal which cannot be generated by a single element.

Historically, the idea of factoring ideals into prime ideals was preceded by Ernst Kummer's introduction of ideal numbers. These are numbers lying in an extension field *E* of *K*. This extension field is now known as the Hilbert class field. By the principal ideal theorem, every prime ideal of *O* generates a principal ideal of the ring of integers of *E*. A generator of this principal ideal is called an ideal number. Kummer used these as a substitute for the failure of unique factorization in cyclotomic fields. These eventually led Richard Dedekind to introduce a forerunner of ideals and to prove unique factorization of ideals.

An ideal which is prime in the ring of integers in one number field may fail to be prime when extended to a larger number field. Consider, for example, the prime numbers. The corresponding ideals *p***Z** are prime ideals of the ring **Z**. However, when this ideal is extended to the Gaussian integers to obtain *p***Z**[*i*], it may or may not be prime. For example, the factorization 2 = (1 + *i*)(1 −*i*) implies that

note that because 1 + *i* = (1 −*i*) ⋅ *i*, the ideals generated by 1 + *i* and 1 −*i* are the same. A complete answer to the question of which ideals remain prime in the Gaussian integers is provided by Fermat's theorem on sums of two squares. It implies that for an odd prime number *p*, *p***Z**[*i*] is a prime ideal if *p* ≡ 3 (mod 4) and is not a prime ideal if *p* ≡ 1 (mod 4). This, together with the observation that the ideal (1 + *i*)**Z**[*i*] is prime, provides a complete description of the prime ideals in the Gaussian integers. 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** (that is, a Galois extension with abelian Galois group).

Unique factorization fails if and only if there are prime ideals that fail to be principal. The object which measures the failure of prime ideals to be principal is called the ideal class group. Defining the ideal class group requires enlarging the set of ideals in a ring of algebraic integers so that they admit a group structure. This is done by generalizing ideals to fractional ideals. A fractional ideal is an additive subgroup *J* of *K* which is closed under multiplication by elements of *O*, meaning that *xJ* ⊆ *J* if *x* ∈ *O*. All ideals of *O* are also fractional ideals. If *I* and *J* are fractional ideals, then the set *IJ* of all products of an element in *I* and an element in *J* is also a fractional ideal. This operation makes the set of non-zero fractional ideals into a group. The group identity is the ideal (1) = *O*, and the inverse of *J* is a (generalized) ideal quotient:

The principal fractional ideals, meaning the ones of the form *Ox* where *x* ∈ *K*^{×}, form a subgroup of the group of all non-zero fractional ideals. The quotient of the group of non-zero fractional ideals by this subgroup is the ideal class group. Two fractional ideals *I* and *J* represent the same element of the ideal class group if and only if there exists an element *x* ∈ *K* such that *xI* = *J*. Therefore, the ideal class group makes two fractional ideals equivalent if one is as close to being principal as the other is. The ideal class group is generally denoted Cl *K*, Cl *O*, or Pic *O* (with the last notation identifying it with the Picard group in algebraic geometry).

The number of elements in the class group is called the **class number** of *K*. The class number of **Q**(√-5) is 2. This means that there are only two ideal classes, the class of principal fractional ideals, and the class of a non-principal fractional ideal such as (2, 1 + √-5).

The ideal class group has another description in terms of divisors. These are formal objects which represent possible factorizations of numbers. The divisor group Div *K* is defined to be the free abelian group generated by the prime ideals of *O*. There is a group homomorphism from *K*^{×}, the non-zero elements of *K* up to multiplication, to Div *K*. Suppose that *x* ∈ *K* satisfies

Then div *x* is defined to be the divisor

The kernel of div is the group of units in *O*, while the cokernel is the ideal class group. In the language of homological algebra, this says that there is an exact sequence of abelian groups (written multiplicatively),

Some number fields, such as **Q**(√2), can be specified as subfields of the real numbers. Others, such as **Q**(√−1), cannot. Abstractly, such a specification corresponds to a field homomorphism *K* → **R** or *K* → **C**. These are called **real embeddings** and **complex embeddings**, respectively.

A real quadratic field **Q**(√*a*), with *a* ∈ **R**, *a* > 0, and *a* not a perfect square, is so-called because it admits two real embeddings but no complex embeddings. These are the field homomorphisms which send √*a* to √*a* and to −√*a*, respectively. Dually, an imaginary quadratic field **Q**(√−*a*) admits no real embeddings but admits a conjugate pair of complex embeddings. One of these embeddings sends √−*a* to √−*a*, while the other sends it to its complex conjugate, −√−*a*.

Conventionally, the number of real embeddings of *K* is denoted *r*_{1}, while the number of conjugate pairs of complex embeddings is denoted *r*_{2}. The **signature** of *K* is the pair (*r*_{1}, *r*_{2}). It is a theorem that *r*_{1} + 2*r*_{2} = *d*, where *d* is the degree of *K*.

Considering all embeddings at once determines a function

This is called the **Minkowski embedding**. The subspace of the codomain fixed by complex conjugation is a real vector space of dimension *d* called Minkowski space. Because the Minkowski embedding is defined by field homomorphisms, multiplication of elements of *K* by an element *x* ∈ *K* corresponds to multiplication by a diagonal matrix in the Minkowski embedding. The dot product on Minkowski space corresponds to the trace form .

The image of *O* under the Minkowski embedding is a *d*-dimensional lattice. If *B* is a basis for this lattice, then det *B*^{T}*B* is the **discriminant** of *O*. The discriminant is denoted Δ or *D*. The covolume of the image of *O* is .

Real and complex embeddings can be put on the same footing as prime ideals by adopting a perspective based on valuations. Consider, for example, the integers. In addition to the usual absolute value function |·| : **Q** → **R**, there are p-adic absolute value functions |·|_{p} : **Q** → **R**, defined for each prime number *p*, which measure divisibility by *p*. Ostrowski's theorem states that these are all possible absolute value functions on **Q** (up to equivalence). Therefore, absolute values are a common language to describe both the real embedding of **Q** and the prime numbers.

A **place** of an algebraic number field is an equivalence class of absolute value functions on *K*. There are two types of places. There is a -adic absolute value for each prime ideal of *O*, and, like the *p*-adic absolute values, it measures divisibility. These are called **finite places**. The other type of place is specified using a real or complex embedding of *K* and the standard absolute value function on **R** or **C**. These are **infinite places**. Because absolute values are unable to distinguish between a complex embedding and its conjugate, a complex embedding and its conjugate determine the same place. Therefore, there are *r*_{1} real places and *r*_{2} complex places. Because places encompass the primes, places are sometimes referred to as **primes**. When this is done, finite places are called **finite primes** and infinite places are called **infinite primes**. If *v* is a valuation corresponding to an absolute value, then one frequently writes to mean that *v* is an infinite place and to mean that it is a finite place.

Considering all the places of the field together produces the adele ring of the number field. The adele ring allows one to simultaneously track all the data available using absolute values. This produces significant advantages in situations where the behavior at one place can affect the behavior at other places, as in the Artin reciprocity law.

There is a geometric analogy for places at infinity which holds on the function fields of curves. For example, let and be a smooth, projective, algebraic curve. The function field has many absolute values, or places, and each corresponds to a point on the curve. If is the projective completion of an affine curve

then the points in

correspond to the places at infinity. Then, the completion of at one of these points gives an analogue of the -adics. For example, if then its function field is isomorphic to where is an indeterminant and the field is the field of fractions of polynomials in . Then, a place at a point measures the order of vanishing or the order of a pole of a fraction of polynomials at the point . For example, if , so on the affine chart this corresponds to the point , the valuation measures the order of vanishing of minus the order of vanishing of at . The function field of the completion at the place is then which is the field of power series in the variable , so an element is of the form

for some . For the place at infinity, this corresponds to the function field which are power series of the form

The integers have only two units, 1 and −1. Other rings of integers may admit more units. The Gaussian integers have four units, the previous two as well as ±*i*. The Eisenstein integers **Z**[exp(2π*i* / 3)] have six units. The integers in real quadratic number fields have infinitely many units. For example, in **Z**[√3], every power of 2 + √3 is a unit, and all these powers are distinct.

In general, the group of units of *O*, denoted *O*^{×}, is a finitely generated abelian group. The fundamental theorem of finitely generated abelian groups therefore implies that it is a direct sum of a torsion part and a free part. Reinterpreting this in the context of a number field, the torsion part consists of the roots of unity that lie in *O*. This group is cyclic. The free part is described by Dirichlet's unit theorem. This theorem says that rank of the free part is *r*_{1} + *r*_{2}− 1. Thus, for example, the only fields for which the rank of the free part is zero are **Q** and the imaginary quadratic fields. 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.^{ [14] }

The free part of the unit group can be studied using the infinite places of *K*. Consider the function

where *v* varies over the infinite places of *K* and |·|_{v} is the absolute value associated with *v*. The function *L* is a homomorphism from *K*^{×} to a real vector space. It can be shown that the image of *O*^{×} is a lattice that spans the hyperplane defined by The covolume of this lattice is the **regulator** of the number field. One of the simplifications made possible by working with the adele ring is that there is a single object, the idele class group, that describes both the quotient by this lattice and the ideal class group.

The Dedekind zeta function of a number field, analogous to the Riemann zeta function is an analytic object which describes the behavior of prime ideals in *K*. When *K* is an abelian extension of **Q**, Dedekind zeta functions are products of Dirichlet L-functions, with there being one factor for each Dirichlet character. The trivial character corresponds to the Riemann zeta function. When *K* is a Galois extension, the Dedekind zeta function is the Artin L-function of the regular representation of the Galois group of *K*, and it has a factorization in terms of irreducible Artin representations of the Galois group.

The zeta function is related to the other invariants described above by the class number formula.

Completing a number field *K* at a place *w* gives a complete field. If the valuation is Archimedean, one obtains **R** or **C**, if it is non-Archimedean and lies over a prime *p* of the rationals, one obtains a finite extension 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. The philosophy behind the study of local fields is largely motivated by geometric methods. In algebraic geometry, it is common to study varieties locally at a point by localizing to a maximal ideal. Global information can then be recovered by gluing together local data. This spirit is adopted in algebraic number theory. Given a prime in the ring of algebraic integers in a number field, it is desirable to study the field locally at that prime. Therefore, one localizes the ring of algebraic integers to that prime and then completes the fraction field much in the spirit of geometry.

One of the classical results in algebraic number theory is that the ideal class group of an algebraic number field *K* is finite. This is a consequence of Minkowski's theorem since there are only finitely many Integral ideals with norm less than a fixed positive integer^{ [15] }^{page 78}. The order of the class group is called the class number, and is often denoted by the letter *h*.

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 number 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*.

In terms of the Legendre symbol, the law of quadratic reciprocity for positive odd primes states

A **reciprocity law** is a generalization of the law of quadratic reciprocity.

There are several different ways to express reciprocity laws. The early reciprocity laws found in the 19th century were usually expressed in terms of a power residue symbol (*p*/*q*) generalizing the quadratic reciprocity symbol, that describes when a prime number is an *n*th power residue modulo another prime, and gave a relation between (*p*/*q*) and (*q*/*p*). Hilbert reformulated the reciprocity laws as saying that a product over *p* of Hilbert symbols (*a*,*b*/*p*), taking values in roots of unity, is equal to 1. Artin's reformulated reciprocity law states that the Artin symbol from ideals (or ideles) to elements of a Galois group is trivial on a certain subgroup. Several more recent generalizations express reciprocity laws using cohomology of groups or representations of adelic groups or algebraic K-groups, and their relationship with the original quadratic reciprocity law can be hard to see.

The **class number formula** relates many important invariants of a number field to a special value of its Dedekind zeta function.

Algebraic number theory interacts with many other mathematical disciplines. It uses tools from homological algebra. Via the analogy of function fields vs. number fields, it relies on techniques and ideas from algebraic geometry. Moreover, the study of higher-dimensional schemes over **Z** instead of number rings is referred to as arithmetic geometry. Algebraic number theory is also used in the study of arithmetic hyperbolic 3-manifolds.

- ↑ Stark, pp. 145–146.
- ↑ Aczel, pp. 14–15.
- ↑ Stark, pp. 44–47.
- ↑ Gauss, Carl Friedrich; Waterhouse, William C. (2018) [1966],
*Disquisitiones Arithmeticae*, Springer, ISBN 978-1-4939-7560-0 - 1 2 Elstrodt, Jürgen (2007), "The Life and Work of Gustav Lejeune Dirichlet (1805–1859)" (PDF),
*Clay Mathematics Proceedings*, retrieved 2007-12-25 - ↑ Kanemitsu, Shigeru; Chaohua Jia (2002),
*Number theoretic methods: future trends*, Springer, pp. 271–4, ISBN 978-1-4020-1080-4 - ↑ Reid, Constance (1996),
*Hilbert*, Springer, ISBN 0-387-94674-8 - ↑ This work established Takagi as Japan's first mathematician of international stature.
- ↑ Hasse, Helmut, "History of Class Field Theory",
*Cassels & Frölich 2010*, pp. 266–279 - ↑ Singh, Simon (1997),
*Fermat's Last Theorem*, ISBN 1-85702-521-0 - ↑ Kolata, Gina (24 June 1993). "At Last, Shout of 'Eureka!' In Age-Old Math Mystery".
*The New York Times*. Retrieved 21 January 2013. - ↑ This notation indicates the ring obtained from
**Z**by adjoining to**Z**the element*i*. - ↑ This notation indicates the ring obtained from
**Z**by adjoining to**Z**the element √-5. - ↑ See proposition VIII.8.6.11 of Neukirch, Schmidt & Wingberg 2000
- ↑ Stein. "A Computational Introduction to Algebraic Number Theory" (PDF).

- Neukirch, Jürgen; Schmidt, Alexander; Wingberg, Kay (2000),
*Cohomology of Number Fields*,*Grundlehren der Mathematischen Wissenschaften*,**323**, Berlin: Springer-Verlag, ISBN 978-3-540-66671-4, MR 1737196, Zbl 0948.11001

- Stein, William (2012),
*Algebraic Number Theory, A Computational Approach*(PDF) - Ireland, Kenneth; Rosen, Michael (2013),
*A classical introduction to modern number theory*,**84**, Springer, doi:10.1007/978-1-4757-2103-4, ISBN 978-1-4757-2103-4 - Stewart, Ian; Tall, David (2015),
*Algebraic Number Theory and Fermat's Last Theorem*, CRC Press, ISBN 978-1-4987-3840-8

- Marcus, Daniel A. (2018),
*Number Fields*(2nd ed.), Springer, ISBN 978-3-319-90233-3

- Cassels, J. W. S.; Fröhlich, Albrecht, eds. (2010) [1967],
*Algebraic number theory*(2nd ed.), London: 9780950273426, MR 0215665 - Fröhlich, Albrecht; Taylor, Martin J. (1993),
*Algebraic number theory*, Cambridge Studies in Advanced Mathematics,**27**, Cambridge University Press, ISBN 0-521-43834-9, MR 1215934 - Lang, Serge (1994),
*Algebraic number theory*, Graduate Texts in Mathematics,**110**(2 ed.), New York: Springer-Verlag, ISBN 978-0-387-94225-4, MR 1282723 - Neukirch, Jürgen (1999).
*Algebraic Number Theory*.*Grundlehren der mathematischen Wissenschaften*.**322**. Berlin: Springer-Verlag. ISBN 978-3-540-65399-8. MR 1697859. Zbl 0956.11021.

- Media related to Algebraic number theory at Wikimedia Commons
- "Algebraic number theory",
*Encyclopedia of Mathematics*, EMS Press, 2001 [1994]

In mathematics, more specifically in ring theory, a **Euclidean domain** is an integral domain that can be endowed with a Euclidean function which allows a suitable generalization of the Euclidean division of the integers. This generalized Euclidean algorithm can be put to many of the same uses as Euclid's original algorithm in the ring of integers: in any Euclidean domain, one can apply the Euclidean algorithm to compute the greatest common divisor of any two elements. In particular, the greatest common divisor of any two elements exists and can be written as a linear combination of them. Also every ideal in a Euclidean domain is principal, which implies a suitable generalization of the fundamental theorem of arithmetic: every Euclidean domain is a unique factorization domain.

In number theory, the **fundamental theorem of arithmetic**, also called the **unique factorization theorem** or the **unique-prime-factorization theorem**, states that every integer greater than 1 either is a prime number itself or can be represented as the product of prime numbers and that, moreover, this representation is unique, up to the order of the factors. For example,

In mathematics, specifically abstract algebra, an **integral domain** is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural setting for studying divisibility. In an integral domain, every nonzero element *a* has the cancellation property, that is, if *a* ≠ 0, an equality *ab* = *ac* implies *b* = *c*.

In mathematics, 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 principal rings. The distinction is that a principal ideal ring may have zero divisors whereas a principal ideal domain cannot.

In number theory, a **Gaussian integer** is a complex number whose real and imaginary parts are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as **Z**[*i*]. This integral domain is a particular case of a commutative ring of quadratic integers. It does not have a total ordering that respects arithmetic.

In mathematics, more specifically in the area of abstract algebra known as ring theory, a **Noetherian ring** is a ring that satisfies the ascending chain condition on left and right ideals; that is, given any increasing sequence of left ideals:

In mathematics, a **unique factorization domain** (**UFD**) is a ring in which a statement analogous to the fundamental theorem of arithmetic holds. Specifically, a UFD is an integral domain in which every non-zero non-unit element can be written as a product of prime elements, uniquely up to order and units.

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. There are at least three other characterizations of Dedekind domains that are sometimes taken as the definition: see below.

In number theory, the **ideal class group** of an algebraic number field *K* is the quotient group *J _{K}*/

The Riemann hypothesis is one of the most important conjectures in mathematics. It is a statement about the zeros of the Riemann zeta function. Various geometrical and arithmetical objects can be described by so-called **global L-functions**, which are formally similar to the Riemann zeta-function. One can then ask the same question about the zeros of these

In mathematics, an **irreducible polynomial** is, roughly speaking, a polynomial that cannot be factored into the product of two non-constant polynomials. The property of irreducibility depends on the nature of the coefficients that are accepted for the possible factors, that is, the field or ring to which the coefficients of the polynomial and its possible factors are supposed to belong. For example, the polynomial *x*^{2} − 2 is a polynomial with integer coefficients, but, as every integer is also a real number, it is also a polynomial with real coefficients. It is irreducible if it is considered as a polynomial with integer coefficients, but it factors as if it is considered as a polynomial with real coefficients. One says that the polynomial *x*^{2} − 2 is irreducible over the integers but not over the reals.

In mathematics, the **ring of integers** of an algebraic number field K is the ring of all integral elements contained in K. An integral element is a root of a monic polynomial with integer coefficients, *x*^{n} + *c*_{n−1}*x*^{n−1} + ... + *c*_{0} . This ring is often denoted by O_{K} or . Since any integer belongs to K and is an integral element of K, the ring **Z** is always a subring of O_{K}.

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. In contexts where fractional ideals and ordinary ring ideals are both under discussion, the latter are sometimes termed * integral ideals* for clarity.

In number theory an **ideal number** is an algebraic integer which represents an ideal in the ring of integers of a number field; the idea was developed by Ernst Kummer, and led to Richard Dedekind's definition of ideals for rings. An ideal in the ring of integers of an algebraic number field is *principal* if it consists of multiples of a single element of the ring, and *nonprincipal* otherwise. By the principal ideal theorem any nonprincipal ideal becomes principal when extended to an ideal of the Hilbert class field. This means that there is an element of the ring of integers of the Hilbert class field, which is an ideal number, such that the original nonprincipal ideal is equal to the collection of all multiples of this ideal number by elements of this ring of integers that lie in the original field's ring of integers.

**Chebotarev's density theorem** in algebraic number theory describes statistically the splitting of primes in a given Galois extension *K* of the field of rational numbers. Generally speaking, a prime integer will factor into several ideal primes in the ring of algebraic integers of *K*. There are only finitely many patterns of splitting that may occur. Although the full description of the splitting of every prime *p* in a general Galois extension is a major unsolved problem, the Chebotarev density theorem says that the frequency of the occurrence of a given pattern, for all primes *p* less than a large integer *N*, tends to a certain limit as *N* goes to infinity. It was proved by Nikolai Chebotaryov in his thesis in 1922, published in.

In mathematics, the **Dedekind zeta function** of an algebraic number field *K*, generally denoted ζ_{K}(*s*), is a generalization of the Riemann zeta function. It can be defined as a Dirichlet series, it has an Euler product expansion, it satisfies a functional equation, it has an analytic continuation to a meromorphic function on the complex plane **C** with only a simple pole at *s* = 1, and its values encode arithmetic data of *K*. The extended Riemann hypothesis states that if *ζ*_{K}(*s*) = 0 and 0 < Re(*s*) < 1, then Re(*s*) = 1/2.

In mathematics, the **discriminant** of an algebraic number field is a numerical invariant that, loosely speaking, measures the size of the algebraic number field. More specifically, it is proportional to the squared volume of the fundamental domain of the ring of integers, and it regulates which primes are ramified.

In mathematics, in the field of abstract algebra, the **structure theorem for finitely generated modules over a principal ideal domain** is a generalization of the fundamental theorem of finitely generated abelian groups and roughly states that finitely generated modules over a principal ideal domain (PID) can be uniquely decomposed in much the same way that integers have a prime factorization. The result provides a simple framework to understand various canonical form results for square matrices over fields.

In number theory, **quadratic integers** are a generalization of the integers to quadratic fields. Quadratic integers are algebraic integers of degree two, that is, solutions of equations of the form

In mathematics, an **algebraic number field** is a finite degree field extension of the field of rational numbers . Thus is a field that contains and has finite dimension when considered as a vector space over .

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