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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 arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777â ...
, a cyclotomic field is a
number field In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a f ...
obtained by adjoining a
complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
root of unity In mathematics, a root of unity, occasionally called a Abraham de Moivre, de Moivre number, is any complex number that yields 1 when exponentiation, raised to some positive integer power . Roots of unity are used in many branches of mathematic ...
to , the
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
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 ration ...
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 a ...
and number theory because of their relation with
Fermat's Last Theorem In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers , , and satisfy the equation for any integer value of greater than 2. The cases and have been k ...
. 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 ...
 ) â€“ and more precisely, because of the failure of
unique factorization In mathematics, a unique factorization domain (UFD) (also sometimes called a factorial ring following the terminology of Bourbaki) is a ring in which a statement analogous to the fundamental theorem of arithmetic holds. Specifically, a UFD is a ...
in their
rings of integers In mathematics, the ring of integers of an algebraic number field K is the ring (mathematics), ring of all algebraic integers contained in K. An algebraic integer is a root of a polynomial, root of a monic polynomial with integer coefficients: x^n+ ...
 â€“ that
Ernst Kummer Ernst Eduard Kummer (29 January 1810 – 14 May 1893) was a German mathematician. Skilled in applied mathematics, Kummer trained German army officers in ballistics; afterwards, he taught for 10 years in a '' gymnasium'', the German equivalent of ...
first introduced the concept of an
ideal number 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 ...
and proved his celebrated
congruences In abstract algebra, a congruence relation (or simply congruence) is an equivalence relation on an algebraic structure (such as a group, ring, or vector space) that is compatible with the structure in the sense that algebraic operations done wi ...
.


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 * E ...
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 primiti ...
: \Phi_n(x) = \!\!\!\prod_\stackrel\!\!\! \left(x-e^\right) = \!\!\!\prod_\stackrel\!\!\! (x-^k) :is
irreducible In philosophy, systems theory, science, and art, emergence occurs when an entity is observed to have properties its parts do not have on their own, properties or behaviors that emerge only when the parts interact in a wider whole. Emergence ...
, 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 Degree may refer to: As a unit of measurement * Degree (angle), a unit of angle measurement ** Degree of geographical latitude ** Degree of geographical longitude * Degree symbol (°), a notation used in science, engineering, and mathematics ...
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 ot ...
. * 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 In abstract algebra, a splitting field of a polynomial with coefficients in a field is the smallest field extension of that field over which the polynomial ''splits'', i.e., decomposes into linear factors. Definition A splitting field of a poly ...
of (or of ) over . * Therefore is a
Galois extension In mathematics, a Galois extension is an algebraic field extension ''E''/''F'' that is normal and separable; or equivalently, ''E''/''F'' is algebraic, and the field fixed by the automorphism group Aut(''E''/''F'') is precisely the base field ...
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 pol ...
\operatorname(\mathbf(\zeta_n)/\mathbf) 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 (\mathbf/n\mathbf)^\times, which consists of the invertible residues modulo , 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 is ...
sends each \sigma \in \operatorname(\mathbf(\zeta_n)/\mathbf) 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 language ...
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 origi ...
of the extension is :: (-1)^\, \frac . * 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 endomorphism ma ...
\operatorname_q \in \operatorname(\mathbf(\zeta_n)/\mathbf) corresponds to the residue of in (\mathbf/n\mathbf)^\times. * The group of roots of unity in has order or , according to whether is even or odd. * The
unit group In algebra, a unit of a ring is an invertible element for the multiplication of the ring. That is, an element of a ring is a unit if there exists in such that vu = uv = 1, where is the multiplicative identity; the element is unique for this ...
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, n ...
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 an abelian group, rank of the group of units in the ring (mathematics), ring of algebraic intege ...
. In particular, is
finite Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marke ...
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 In group theory, a branch of mathematics, given a group ''G'' under a binary operation âˆ—, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation âˆ—. More precisely, ''H'' is a subgroup ...
of . * The
Kronecker–Weber theorem In algebraic number theory, it can be shown that every cyclotomic field is an abelian extension of the rational number field Q, having Galois group of the form (\mathbb Z/n\mathbb Z)^\times. The Kronecker–Weber theorem provides a partial conve ...
states that every
finite Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marke ...
abelian extension In abstract algebra, an abelian extension is a Galois extension whose Galois group is abelian. When the Galois group is also cyclic, the extension is also called a cyclic extension. Going in the other direction, a Galois extension is called solvabl ...
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 Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
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 A power of two is a number of the form where is an integer, that is, the result of exponentiation with number two as the base and integer  as the exponent. In a context where only integers are considered, is restricted to non-negative ...
; * n=2^a p_1 \cdots p_r for some integers and
Fermat prime In mathematics, a Fermat number, named after Pierre de Fermat, who first studied them, is a positive integer of the form :F_ = 2^ + 1, where ''n'' is a non-negative integer. The first few Fermat numbers are: : 3, 5, 17, 257, 65537, 4294967 ...
s p_1,\ldots,p_r. (A Fermat prime is an odd prime such that is a power of 2. The known Fermat primes are 3, 5, 17, 257,
65537 65537 is the integer after 65536 and before 65538. In mathematics 65537 is the largest known prime number of the form 2^ +1 (n = 4). Therefore, a regular polygon with 65537 sides is constructible with compass and unmarked straightedge. Johann ...
, and it is likely that there are no others.)


Small examples

* and : The equations \zeta_3 = \tfrac and \zeta_6 = \tfrac 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 In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers , , and satisfy the equation for any integer value of greater than 2. The cases and have been k ...
is to factor the binomial , where is an odd prime, appearing in one side of Fermat's equation : x^n + y^n = z^n as follows: : x^n + y^n = (x + y)(x + \zeta y)\cdots (x + \zeta^ y) Here and are ordinary integers, whereas the factors are
algebraic integer In algebraic number theory, an algebraic integer is a complex number which is integral over the integers. That is, an algebraic integer is a complex root of some monic polynomial (a polynomial whose leading coefficient is 1) whose coefficients ...
s in the cyclotomic field . If
unique factorization In mathematics, a unique factorization domain (UFD) (also sometimes called a factorial ring following the terminology of Bourbaki) is a ring in which a statement analogous to the fundamental theorem of arithmetic holds. Specifically, a UFD is a ...
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 prime In number theory, a regular prime is a special kind of prime number, defined by Ernst Kummer in 1850 to prove certain cases of Fermat's Last Theorem. Regular primes may be defined via the divisibility of either class numbers or of Bernoulli num ...
s'') 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 the ea ...
and by Kubota and Leopoldt in their theory of ''p''-adic zeta functions.


List of class numbers of cyclotomic fields

, or or for the h-part (for prime ''n'')


See also

*
Kronecker–Weber theorem In algebraic number theory, it can be shown that every cyclotomic field is an abelian extension of the rational number field Q, having Galois group of the form (\mathbb Z/n\mathbb Z)^\times. The Kronecker–Weber theorem provides a partial conve ...
*
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 primiti ...


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 John William Scott "Ian" Cassels, FRS (11 July 1922 – 27 July 2015) was a British mathematician. Biography Cassels was educated at Neville's Cross Council School in Durham and George Heriot's School in Edinburgh. He went on to study a ...
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'' value, a mea ...
(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 reference ...
, 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 Karl Cooper Rubin (born January 27, 1956) is an American mathematician at University of California, Irvine as Thorp Professor of Mathematics. Between 1997 and 2006, he was a professor at Stanford, and before that worked at Ohio State University b ...
.
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 s ...
, 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 *