# Cyclotomic field

In number theory, a cyclotomic field is a number field obtained by adjoining a complex root of unity to Q, 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 n) – 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 n ≥ 1, let ζn = ei/nC; this is a primitive nth root of unity. Then the nth cyclotomic field is the extension Qn) of Q generated by ζn.

## Properties

${\displaystyle \Phi _{n}(x)=\!\!\!\prod _{\stackrel {1\leq k\leq n}{\gcd(k,n)=1}}\!\!\!\left(x-e^{2\pi ik/n}\right)=\!\!\!\prod _{\stackrel {1\leq k\leq n}{\gcd(k,n)=1}}\!\!\!(x-{\zeta _{n}}^{k})}$

is irreducible, so it is the minimal polynomial of ζn over Q.

• The conjugates of ζn in C are therefore the other primitive nth roots of unity: ζnk for 1 ≤ kn with gcd(k,n) = 1.
• The degree of Qn) is therefore [Qn):Q] = deg Φn = φ(n), where φ is Euler's totient function.
• The roots of xn − 1 are the powers of ζn, so Qn) is the splitting field of xn − 1 (or of Φ(x)) over Q.
• Therefore Qn) is a Galois extension of Q.
• The Galois group ${\displaystyle \operatorname {Gal} (\mathbf {Q} (\zeta _{n})/\mathbf {Q} )}$ is naturally isomorphic to the multiplicative group ${\displaystyle (\mathbf {Z} /n\mathbf {Z} )^{\times }}$, which consists of the invertible residues modulo n, which are the residues a mod n with 1 ≤ an and gcd(a,n) = 1. The isomorphism sends each ${\displaystyle \sigma \in \operatorname {Gal} (\mathbf {Q} (\zeta _{n})/\mathbf {Q} )}$ to a mod n, where a is an integer such that σ(ζn) = ζna.
• The ring of integers of Qn) is Zn].
• For n > 2, the discriminant of the extension Qn)/Q is[1]
${\displaystyle (-1)^{\varphi (n)/2}{\frac {n^{\varphi (n)}}{\displaystyle \prod _{p|n}p^{\varphi (n)/(p-1)}}}.}$
• In particular, Qn)/Q is unramified above every prime not dividing n.
• If n is a power of a prime p, then Qn)/Q is totally ramified above p.
• If q is a prime not dividing n, then the Frobenius element ${\displaystyle \operatorname {Frob} _{q}\in \operatorname {Gal} (\mathbf {Q} (\zeta _{n})/\mathbf {Q} )}$ corresponds to the residue of q in ${\displaystyle (\mathbf {Z} /n\mathbf {Z} )^{\times }}$.
• The group of roots of unity in Qn) has order n or 2n, according to whether n is even or odd.
• The unit group Zn]× is a finitely generated abelian group of rank φ(n)/2 – 1, for any n > 2, by the Dirichlet unit theorem. In particular, Zn]× is finite only for n ∈ {1,2,3,4,6}. The torsion subgroup of Zn]× is the group of roots of unity in Qn), which was described in the previous item. Cyclotomic units form an explicit finite-index subgroup of Zn]×.
• The Kronecker–Weber theorem states that every finite abelian extension of Q in C is contained in Qn) for some n. Equivalently, the union of all the cyclotomic fields Qn) is the maximal abelian extension Qab of Q.

## Relation with regular polygons

Gauss made early inroads in the theory of cyclotomic fields, in connection with the problem of constructing a regular n-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 n ≥ 3, the following are equivalent:

• a regular n-gon is constructible;
• there is a sequence of fields, starting with Q and ending with Qn), such that each is a quadratic extension of the previous field;
• φ(n) is a power of 2;
• ${\displaystyle n=2^{a}p_{1}\cdots p_{r}}$ for some integers a, r ≥ 0 and Fermat primes ${\displaystyle p_{1},\ldots ,p_{r}}$. (A Fermat prime is an odd prime p such that p − 1 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

• n = 3 and n = 6: The equations ${\displaystyle \zeta _{3}={\tfrac {-1+{\sqrt {-3}}}{2}}}$ and ${\displaystyle \zeta _{6}={\tfrac {1+{\sqrt {-3}}}{2}}}$ show that Q3) = Q6) = Q(-3), which is a quadratic extension of Q. Correspondingly, a regular 3-gon and a regular 6-gon are constructible.
• n = 4: Similarly, ζ4 = i, so Q4) = Q(i), and a regular 4-gon is constructible.
• n = 5: The field Q5) is not a quadratic extension of Q, but it is a quadratic extension of the quadratic extension Q(5), 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 xn + yn, where n is an odd prime, appearing in one side of Fermat's equation

${\displaystyle x^{n}+y^{n}=z^{n}}$

as follows:

${\displaystyle x^{n}+y^{n}=(x+y)(x+\zeta y)\cdots (x+\zeta ^{n-1}y)}$

Here x and y are ordinary integers, whereas the factors are algebraic integers in the cyclotomic field Q(ζn). If unique factorization holds in the cyclotomic integers Z[ζn] , 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 n = 4 and Euler's proof for n = 3 can be recast in these terms. The complete list of n for which Q(ζn) has unique factorization is[2]

• 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 Z[ζn], measured the failure of unique factorization via the class number hn and proved that if hp is not divisible by a prime p (such p are called regular primes) then Fermat's theorem is true for the exponent n = p. Furthermore, he gave a criterion to determine which primes are regular, and established Fermat's theorem for all prime exponents p 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

(sequence A061653 in the OEIS), or or for the ${\displaystyle h}$-part (for prime n)

• 1-22: 1
• 23: 3
• 24-28: 1
• 29: 8
• 30: 1
• 31: 9
• 32-36: 1
• 37: 37
• 38: 1
• 39: 2
• 40: 1
• 41: 121
• 42: 1
• 43: 211
• 44: 1
• 45: 1
• 46: 3
• 47: 695
• 48: 1
• 49: 43
• 50: 1
• 51: 5
• 52: 3
• 53: 4889
• 54: 1
• 55: 10
• 56: 2
• 57: 9
• 58: 8
• 59: 41241
• 60: 1
• 61: 76301
• 62: 9
• 63: 7
• 64: 17
• 65: 64
• 66: 1
• 67: 853513
• 68: 8
• 69: 69
• 70: 1
• 71: 3882809
• 72: 3
• 73: 11957417
• 74: 37
• 75: 11
• 76: 19
• 77: 1280
• 78: 2
• 79: 100146415
• 80: 5
• 81: 2593
• 82: 121
• 83: 838216959
• 84: 1
• 85: 6205
• 86: 211
• 87: 1536
• 88: 55
• 89: 13379363737
• 90: 1
• 91: 53872
• 92: 201
• 93: 6795
• 94: 695
• 95: 107692
• 96: 9
• 97: 411322824001
• 98: 43
• 99: 2883
• 100: 55
• 101: 3547404378125
• 102: 5
• 103: 9069094643165
• 104: 351
• 105: 13
• 106: 4889
• 107: 63434933542623
• 108: 19
• 109: 161784800122409
• 110: 10
• 111: 480852
• 112: 468
• 113: 1612072001362952
• 114: 9
• 115: 44697909
• 116: 10752
• 117: 132678
• 118: 41241
• 119: 1238459625
• 120: 4
• 121: 12188792628211
• 122: 76301
• 123: 8425472
• 124: 45756
• 125: 57708445601
• 126: 7
• 127: 2604529186263992195
• 128: 359057
• 129: 37821539
• 130: 64
• 131: 28496379729272136525
• 132: 11
• 133: 157577452812
• 134: 853513
• 135: 75961
• 136: 111744
• 137: 646901570175200968153
• 138: 69
• 139: 1753848916484925681747
• 140: 39
• 141: 1257700495
• 142: 3882809
• 143: 36027143124175
• 144: 507
• 145: 1467250393088
• 146: 11957417
• 147: 5874617
• 148: 4827501
• 149: 687887859687174720123201
• 150: 11
• 151: 2333546653547742584439257
• 152: 1666737
• 153: 2416282880
• 154: 1280
• 155: 84473643916800
• 156: 156
• 157: 56234327700401832767069245
• 158: 100146415
• 159: 223233182255
• 160: 31365

## References

1. ^ Washington 1997, Proposition 2.7.
2. ^ Washington 1997, Theorem 11.1.

### 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
• Washington, Lawrence C. (1997), Introduction to Cyclotomic Fields, Graduate Texts in Mathematics, vol. 83 (2 ed.), New York: Springer-Verlag, doi:10.1007/978-1-4612-1934-7, ISBN 0-387-94762-0, MR 1421575
• 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. ISBN 0-387-96671-4