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 ===
The beginnings of algebraic number theory
can be traced to Diophantine equations, 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:
A
=
x
+
y
{\displaystyle A=x+y\ }
B
=
x
2
+
y
2
.
{\displaystyle B=x^{2}+y^{2}.\ }
Diophantine equations have been studied for
thousands of years. For example, the solutions
to the quadratic Diophantine equation x2 +
y2 = z2 are given by the Pythagorean triples,
originally solved by the Babylonians (c. 1800
BC). Solutions to linear Diophantine equations,
such as 26x + 65y = 13, may be found using
the Euclidean algorithm (c. 5th century BC).Diophantus'
major work was the Arithmetica, of which only
a portion has survived.
=== Fermat ===
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.
=== Gauss ===
One of the founding works of algebraic number
theory, the Disquisitiones Arithmeticae (Latin:
Arithmetical Investigations) is a textbook
of number theory written in Latin by Carl
Friedrich Gauss in 1798 when Gauss was 21
and first published in 1801 when he was 24.
In this book Gauss brings together results
in number theory obtained by mathematicians
such as Fermat, Euler, Lagrange and Legendre
and adds important new results of his own.
Before the Disquisitiones was published, number
theory consisted of a collection of isolated
theorems and conjectures. Gauss brought the
work of his predecessors together with his
own original work into a systematic framework,
filled in gaps, corrected unsound proofs,
and extended the subject in numerous ways.
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.
=== Dirichlet ===
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. 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.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. 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.
=== Dedekind ===
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.
=== Hilbert ===
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. 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.
=== Artin ===
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. 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.
=== Modern theory ===
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" 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
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.
== Basic notions ==
=== 
Failure of unique factorization ===
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
6
=
2
⋅
3
=
(
−
2
)
⋅
(
−
3
)
.
{\displaystyle 6=2\cdot 3=(-2)\cdot (-3).}
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], the numbers
1 + 2i 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
5
=
(
1
+
2
i
)
(
1
−
2
i
)
=
(
2
+
i
)
(
2
−
i
)
,
{\displaystyle 5=(1+2i)(1-2i)=(2+i)(2-i),}
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]. 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,
9
=
3
2
=
(
2
+
−
5
)
(
2
−
−
5
)
.
{\displaystyle 9=3^{2}=(2+{\sqrt {-5}})(2-{\sqrt
{-5}}).}
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 3a + 3b√-5.
Similarly, 2 + √-5 and 2 - √-5 divide
the product 32, 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.
=== Factorization into prime ideals ===
If I is an ideal in O, then there is always
a factorization
I
=
p
1
e
1
⋯
p
t
e
t
,
{\displaystyle I={\mathfrak {p}}_{1}^{e_{1}}\cdots
{\mathfrak {p}}_{t}^{e_{t}},}
where each
p
i
{\displaystyle {\mathfrak {p}}_{i}}
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 pZ are prime ideals of the ring Z.
However, when this ideal is extended to the
Gaussian integers to get pZ[i], it may or
may not be prime. For example, the factorization
2 = (1 + i)(1 − i) implies that
2
Z
[
i
]
=
(
1
+
i
)
Z
[
i
]
⋅
(
1
−
i
)
Z
[
i
]
=
(
(
1
+
i
)
Z
[
i
]
)
2
;
{\displaystyle 2\mathbf {Z} [i]=(1+i)\mathbf
{Z} [i]\cdot (1-i)\mathbf {Z} [i]=((1+i)\mathbf
{Z} [i])^{2};}
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, pZ[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 (i.e. a Galois extension with abelian Galois
group).
=== Ideal class 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, J−1 = (O
: J) = { x ∈ K : xJ ⊆ O }.
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
(
x
)
=
p
1
e
1
⋯
p
t
e
t
.
{\displaystyle (x)={\mathfrak {p}}_{1}^{e_{1}}\cdots
{\mathfrak {p}}_{t}^{e_{t}}.}
Then div x is defined to be the divisor
div
⁡
x
=
∑
i
=
1
t
e
i
[
p
i
]
.
{\displaystyle \operatorname {div} x=\sum
_{i=1}^{t}e_{i}[{\mathfrak {p}}_{i}].}
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),
1
→
O
×
→
K
×
→
div
Div
⁡
K
→
Cl
⁡
K
→
1.
{\displaystyle 1\to O^{\times }\to K^{\times
}{\xrightarrow {\text{div}}}\operatorname
{Div} K\to \operatorname {Cl} K\to 1.}
=== Real and complex embeddings ===
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 r1, while the number of conjugate
pairs of complex embeddings is denoted r2.
The signature of K is the pair (r1, r2). It
is a theorem that r1 + 2r2 = d, where d is
the degree of K.
Considering all embeddings at once determines
a function
M
:
K
→
R
r
1
⊕
C
2
r
2
.
{\displaystyle M\colon K\to \mathbf {R} ^{r_{1}}\oplus
\mathbf {C} ^{2r_{2}}.}
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
⟨
x
,
y
⟩
=
Tr
⁡
(
x
y
)
{\displaystyle \langle x,y\rangle =\operatorname
{Tr} (xy)}
.
The image of O under the Minkowski embedding
is a d-dimensional lattice. If B is a basis
for this lattice, then det BTB is the discriminant
of O. The discriminant is denoted Δ or D.
The covolume of the image of O is
|
Δ
|
{\displaystyle {\sqrt {|\Delta |}}}
.
=== Places ===
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
p
{\displaystyle {\mathfrak {p}}}
-adic absolute value for each prime ideal
p
{\displaystyle {\mathfrak {p}}}
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 r1 real
places and r2 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
v
∣
∞
{\displaystyle v\mid \infty }
to mean that v is an infinite place and
v
∤
∞
{\displaystyle v\nmid \infty }
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.
=== Units ===
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(2πi / 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 r1 + r2
− 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.The free part
of the unit group can be studied using the
infinite places of K. Consider the function
L
:
K
×
→
R
r
1
+
r
2
{\displaystyle L\colon K^{\times }\to \mathbf
{R} ^{r_{1}+r_{2}}}
defined by
L
(
x
)
=
(
log
⁡
|
x
|
v
)
v
,
{\displaystyle L(x)=(\log |x|_{v})_{v},}
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
x
1
+
⋯
+
x
r
1
+
r
2
=
0
{\displaystyle x_{1}+\cdots +x_{r_{1}+r_{2}}=0}
. 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.
=== Zeta function ===
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.
=== Local fields ===
Completing a number field K 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. 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.
== 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 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 r1 + r2 − 1 whose
torsion consists of the roots of unity in
O.
=== Reciprocity laws ===
In terms of the Legendre symbol, the law of
quadratic reciprocity for positive odd primes
states
(
p
q
)
(
q
p
)
=
(
−
1
)
p
−
1
2
q
−
1
2
.
{\displaystyle \left({\frac {p}{q}}\right)\left({\frac
{q}{p}}\right)=(-1)^{{\frac {p-1}{2}}{\frac
{q-1}{2}}}.}
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 nth 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.
=== Class number formula ===
The class number formula relates many important
invariants of a number field to a special
value of its Dedekind zeta function.
== Related areas ==
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.
== See also ==
Tamagawa number
Arithmetic algebraic geometry
Kummer theory
== Notes ==
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
== Further reading ==
=== 
Introductory texts ===
Stein, William (2012). Algebraic Number Theory,
A Computational Approach. Retrieved from https://wstein.org/books/ant/ant.pdf
Ireland, Kenneth and Rosen, Michael (2013).
A classical introduction to modern number
theory (Vol. 84). Springer Science & Business
Media. doi:10.1007/978-1-4757-2103-4
Stewart, Ian and Tall, David (2015). Algebraic
number theory and Fermat's last theorem. CRC
Press.
=== Intermediate texts ===
Marcus, Daniel A. (1977). Number fields (Vol.
8). New York: Springer.
=== Graduate level accounts ===
Cassels, J. W. S.; Fröhlich, Albrecht, eds.
(1967), Algebraic number theory, London: Academic
Press, 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.
== External links ==
Media related to Algebraic number theory at
Wikimedia Commons
Hazewinkel, Michiel, ed. (2001) [1994], "Algebraic
number theory", Encyclopedia of Mathematics,
Springer Science+Business Media B.V. / Kluwer
Academic Publishers, ISBN 978-1-55608-010-4
