In
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Ma ...
, a cyclotomic field is a
number field obtained by
adjoining a
complex root of unity
In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that yields 1 when raised to some positive integer power . Roots of unity are used in many branches of mathematics, and are especially important i ...
to , the
field of
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
s.
Cyclotomic fields played a crucial role in the development of modern
algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary ...
and number theory because of their relation with
Fermat's Last Theorem. It was in the process of his deep investigations of the arithmetic of these fields (for
prime
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
) – and more precisely, because of the failure of
unique factorization in their
rings of integers – that
Ernst Kummer first introduced the concept of an
ideal number and proved his celebrated
congruences.
Definition
For , let ; this is a
primitive th root of unity. Then the th cyclotomic field is the
extension
Extension, extend or extended may refer to:
Mathematics
Logic or set theory
* Axiom of extensionality
* Extensible cardinal
* Extension (model theory)
* Extension (predicate logic), the set of tuples of values that satisfy the predicate
* Ext ...
of generated by .
Properties
* The th
cyclotomic polynomial
In mathematics, the ''n''th cyclotomic polynomial, for any positive integer ''n'', is the unique irreducible polynomial with integer coefficients that is a divisor of x^n-1 and is not a divisor of x^k-1 for any Its roots are all ''n''th primitiv ...
:
:is
irreducible, so it is the
minimal polynomial of over .
* The
conjugates of in are therefore the other primitive th roots of unity: for with .
* The
degree of is therefore , where is
Euler's totient function
In number theory, Euler's totient function counts the positive integers up to a given integer that are relatively prime to . It is written using the Greek letter phi as \varphi(n) or \phi(n), and may also be called Euler's phi function. In ...
.
* The
roots
A root is the part of a plant, generally underground, that anchors the plant body, and absorbs and stores water and nutrients.
Root or roots may also refer to:
Art, entertainment, and media
* ''The Root'' (magazine), an online magazine focusing ...
of are the powers of , so is the
splitting field of (or of ) over .
* Therefore is a
Galois extension of .
* The
Galois group
In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the po ...
is
naturally isomorphic
In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natur ...
to
the multiplicative group , which consists of the invertible residues
modulo
In computing, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another (called the '' modulus'' of the operation).
Given two positive numbers and , modulo (often abbreviated as ) is ...
, which are the residues with and . The
isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
sends each
to , where is an
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
such that .
* The
ring of integers
In mathematics, the ring of integers of an algebraic number field K is the ring of all algebraic integers contained in K. An algebraic integer is a root of a monic polynomial with integer coefficients: x^n+c_x^+\cdots+c_0. This ring is often deno ...
of is .
* For , the
discriminant
In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the roots without computing them. More precisely, it is a polynomial function of the coefficients of the orig ...
of the extension is
::
* In particular, is
unramified
In geometry, ramification is 'branching out', in the way that the square root function, for complex numbers, can be seen to have two ''branches'' differing in sign. The term is also used from the opposite perspective (branches coming together) as ...
above every prime not dividing .
* If is a power of a prime , then is totally ramified above .
* If is a prime not dividing , then the
Frobenius element
In commutative algebra and field theory, the Frobenius endomorphism (after Ferdinand Georg Frobenius) is a special endomorphism of commutative rings with prime characteristic , an important class which includes finite fields. The endomorphi ...
corresponds to the residue of in
.
* The group of roots of unity in has order or , according to whether is even or odd.
* The
unit group is a
finitely generated abelian group
In abstract algebra, an abelian group (G,+) is called finitely generated if there exist finitely many elements x_1,\dots,x_s in G such that every x in G can be written in the form x = n_1x_1 + n_2x_2 + \cdots + n_sx_s for some integers n_1,\dots, ...
of rank , for any , by the
Dirichlet unit theorem
In mathematics, Dirichlet's unit theorem is a basic result in algebraic number theory due to Peter Gustav Lejeune Dirichlet. It determines the rank of the group of units in the ring of algebraic integers of a number field . The regulator is a pos ...
. In particular, is
finite only for . The
torsion subgroup In the theory of abelian groups, the torsion subgroup ''AT'' of an abelian group ''A'' is the subgroup of ''A'' consisting of all elements that have finite order (the torsion elements of ''A''). An abelian group ''A'' is called a torsion group (or ...
of is the group of roots of unity in , which was described in the previous item.
Cyclotomic units form an explicit finite-
index
Index (or its plural form indices) may refer to:
Arts, entertainment, and media Fictional entities
* Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index''
* The Index, an item on a Halo megastru ...
subgroup of .
* The
Kronecker–Weber theorem states that every
finite abelian extension of in is contained in for some . Equivalently, the union of all the cyclotomic fields is the
maximal abelian extension of .
Relation with regular polygons
Gauss made early inroads in the theory of cyclotomic fields, in connection with the problem of
constructing a
regular -gon with a
compass and straightedge
In geometry, straightedge-and-compass construction – also known as ruler-and-compass construction, Euclidean construction, or classical construction – is the construction of lengths, angles, and other geometric figures using only an ideali ...
. His surprising result that had escaped his predecessors was that a regular
17-gon could be so constructed. More generally, for any integer , the following are equivalent:
* a regular -gon is constructible;
* there is a sequence of fields, starting with and ending with , such that each is a
quadratic extension
In mathematics, particularly in algebra, a field extension is a pair of fields E\subseteq F, such that the operations of ''E'' are those of ''F'' restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' is a subfield of ...
of the previous field;
* is a
power of 2;
*
for some integers and
Fermat primes
. (A Fermat prime is an odd prime such that is a power of 2. The known Fermat primes are
3,
5,
17,
257,
65537, and it is likely that there are no others.)
Small examples
* and : The equations
and
show that , which is a quadratic extension of . Correspondingly, a regular 3-gon and a regular 6-gon are constructible.
* : Similarly, , so , and a regular 4-gon is constructible.
* : The field is not a quadratic extension of , but it is a quadratic extension of the quadratic extension , so a regular 5-gon is constructible.
Relation with Fermat's Last Theorem
A natural approach to proving
Fermat's Last Theorem is to factor the binomial ,
where is an odd prime, appearing in one side of Fermat's equation
:
as follows:
:
Here and are ordinary integers, whereas the factors are
algebraic integers in the cyclotomic field . If
unique factorization holds in the cyclotomic integers , then it can be used to rule out the existence of nontrivial solutions to Fermat's equation.
Several attempts to tackle Fermat's Last Theorem proceeded along these lines, and both Fermat's proof for and Euler's proof for can be recast in these terms. The complete list of for which has unique factorization is
* 1 through 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 40, 42, 44, 45, 48, 50, 54, 60, 66, 70, 84, 90.
Kummer found a way to deal with the failure of unique factorization. He introduced a replacement for the prime numbers in the cyclotomic integers , measured the failure of unique factorization via the
class number and proved that if is not divisible by a prime (such are called ''
regular primes'') then Fermat's theorem is true for the exponent . Furthermore, he
gave a criterion to determine which primes are regular, and established Fermat's theorem for all prime exponents less than 100, except for the ''irregular primes''
37,
59, and
67. Kummer's work on the congruences for the class numbers of cyclotomic fields was generalized in the twentieth century by
Iwasawa in
Iwasawa theory
In number theory, Iwasawa theory is the study of objects of arithmetic interest over infinite towers of number fields. It began as a Galois module theory of ideal class groups, initiated by (), as part of the theory of cyclotomic fields. In th ...
and by Kubota and Leopoldt in their theory of
''p''-adic zeta functions.
List of class numbers of cyclotomic fields
, or or for the
-part (for prime ''n'')
See also
*
Kronecker–Weber theorem
*
Cyclotomic polynomial
In mathematics, the ''n''th cyclotomic polynomial, for any positive integer ''n'', is the unique irreducible polynomial with integer coefficients that is a divisor of x^n-1 and is not a divisor of x^k-1 for any Its roots are all ''n''th primitiv ...
References
Sources
*
Bryan Birch
Bryan John Birch FRS (born 25 September 1931) is a British mathematician. His name has been given to the Birch and Swinnerton-Dyer conjecture.
Biography
Bryan John Birch was born in Burton-on-Trent, the son of Arthur Jack and Mary Edith Birch. ...
, "Cyclotomic fields and Kummer extensions", in
J.W.S. Cassels and
A. Frohlich
A is the first letter of the Latin and English alphabet.
A may also refer to:
Science and technology Quantities and units
* ''a'', a measure for the attraction between particles in the Van der Waals equation
* ''A'' value, a measure of ...
(edd), ''Algebraic number theory'',
Academic Press
Academic Press (AP) is an academic book publisher founded in 1941. It was acquired by Harcourt, Brace & World in 1969. Reed Elsevier bought Harcourt in 2000, and Academic Press is now an imprint of Elsevier.
Academic Press publishes refer ...
, 1973. Chap.III, pp. 45–93.
* Daniel A. Marcus, ''Number Fields'', first edition, Springer-Verlag, 1977
*
*
Serge Lang
Serge Lang (; May 19, 1927 – September 12, 2005) was a French-American mathematician and activist who taught at Yale University for most of his career. He is known for his work in number theory and for his mathematics textbooks, including the i ...
, ''Cyclotomic Fields I and II'', Combined second edition. With an appendix by
Karl Rubin.
Graduate Texts in Mathematics
Graduate Texts in Mathematics (GTM) (ISSN 0072-5285) is a series of graduate-level textbooks in mathematics published by Springer-Verlag. The books in this series, like the other Springer-Verlag mathematics series, are yellow books of a standard ...
, 121. Springer-Verlag, New York, 1990.
Further reading
*
*
* {{springer, title=Cyclotomic field, id=p/c027570
* On the Ring of Integers of Real Cyclotomic Fields. Koji Yamagata and Masakazu Yamagishi: Proc,Japan Academy, 92. Ser a (2016)
Algebraic number theory
*