Cyclotomic field
In number theory, a cyclotomic field is a number field obtained by adjoining a complex root of unity to , the field of rational numbers. Cyclotomic fields played a crucial role in the development of modern algebra 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 ) – 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  of  generated by . Properties
 The th cyclotomic polynomial 

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. The roots of  are the powers of , so  is the splitting field of  (or of ) over . Therefore  is a Galois extension of . The Galois group  is naturally isomorphic to the multiplicative group , which consists of the invertible residues modulo , which are the residues  with  and . The isomorphism sends each  to , where  is an integer such that . The ring of integers of  is . For , the discriminant of the extension  is
 
 In particular,  is unramified 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  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 of rank , for any , by the Dirichlet unit theorem. In particular,  is finite only for }. The torsion subgroup of  is the group of roots of unity in , which was described in the previous item. Cyclotomic units form an explicit finite-index 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. 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 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 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

References

Sources
 Bryan Birch, "Cyclotomic fields and Kummer extensions", in J.W.S. Cassels and A. Frohlich (edd), Algebraic number theory, Academic Press, 1973. Chap.III, pp. 45–93. Daniel A. Marcus, Number Fields, first edition, Springer-Verlag, 1977
 
 Serge Lang, Cyclotomic Fields I and II, Combined second edition. With an appendix by Karl Rubin. Graduate Texts in Mathematics, 121. Springer-Verlag, New York, 1990. Further reading
 
 
 
 On the Ring of Integers of Real Cyclotomic Fields. Koji Yamagata and Masakazu Yamagishi: Proc,Japan Academy, 92. Ser a (2016)
Algebraic number theory