Cyclotomic field

In algebraic number theory, a cyclotomic field is a number field obtained by adjoining a complex root of unity to $\mathbb {Q}$ , the field of rational numbers.^[1]

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 $n\geq 1$ , let

\zeta _{n}=\left\{e^{\frac {2k\pi i}{n}}\;|n,k\in \mathbb {N} \;\gcd(n,k)=1\right\}

.

This is a primitive $n$ th root of unity. Then the $n$ th cyclotomic field is the field extension $\mathbb {Q} (\zeta _{n})$ of $\mathbb {Q}$ generated by $\zeta _{n}$ .

Properties

The $n$ th cyclotomic polynomial

\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

\zeta _{n}

over

\mathbb {Q}

.

The conjugates of $\zeta _{n}$ in $\mathbb {C}$ are therefore the other primitive $n$ th roots of unity: $\zeta _{n}^{k}$ for $1\leq k\leq n$ with $\gcd(k,n)=1$ .

The degree of $\mathbb {Q} (\zeta _{n})$ is therefore $[\mathbb {Q} (\zeta _{n}):\mathbb {Q} ]=\deg \Phi _{n}=\varphi (n)$ , where $\varphi$ is Euler's totient function.

The roots of $x^{n}-1$ are the powers of $\zeta _{n}$ , so $\mathbb {Q} (\zeta _{n})$ is the splitting field of $x^{n}-1$ (or of $\Phi _{n}$ ) over $\mathbb {Q}$ . It follows that $\mathbb {Q} (\zeta _{n})$ is a Galois extension of $\mathbb {Q}$ .

The Galois group $\operatorname {Gal} (\mathbb {Q} (\zeta _{n})/\mathbb {Q} )$ is naturally isomorphic to the multiplicative group $(\mathbb {Z} /n\mathbb {Z} )^{\times }$ , which consists of the invertible residues modulo $n$ , which are the residues $a$ mod $n$ with $1\leq a\leq n$ and $\gcd(a,n)=1$ . The isomorphism sends each $\sigma \in \operatorname {Gal} (\mathbb {Q} (\zeta _{n})/\mathbb {Q} )$ to $a$ mod $n$ , where $a$ is an integer such that $\sigma (\zeta _{n})=\zeta _{n}^{a}$ .

The ring of integers of $\mathbb {Q} (\zeta _{n})$ is $\mathbb {Z} [\zeta _{n}]$ .

For $n>2$ , the discriminant of the extension $\mathbb {Q} (\zeta _{n})/\mathbb {Q}$ is^[2]

(-1)^{\varphi (n)/2}\,{\frac {n^{\varphi (n)}}{\displaystyle \prod _{p|n}p^{\varphi (n)/(p-1)}}}.

In particular, $\mathbb {Q} (\zeta _{n})/\mathbb {Q}$ is unramified above every prime not dividing $n$ .

If $n$ is a power of a prime $p$ , then $\mathbb {Q} (\zeta _{n})/\mathbb {Q}$ is totally ramified above $p$ .

If $q$ is a prime not dividing $n$ , then the Frobenius element $\operatorname {Frob} _{q}\in \operatorname {Gal} (\mathbb {Q} (\zeta _{n})/\mathbb {Q} )$ corresponds to the residue of $q$ in $(\mathbb {Z} /n\mathbb {Z} )^{\times }$ .

The group of roots of unity in $\mathbb {Q} (\zeta _{n})$ has order $n$ or $2n$ , according to whether $n$ is even or odd.

The unit group $\mathbb {Z} [\zeta _{n}]^{\times }$ is a finitely generated abelian group of rank $\varphi (n)/2-1$ , for any $n>2$ , by the Dirichlet unit theorem. In particular, $\mathbb {Z} [\zeta _{n}]^{\times }$ is finite only for $n\in \{1,2,3,4,6\}$ . The torsion subgroup of $\mathbb {Z} [\zeta _{n}]^{\times }$ is the group of roots of unity in $\mathbb {Q} (\zeta _{n})$ , which was described in the previous item. Cyclotomic units form an explicit finite-index subgroup of $\mathbb {Z} [\zeta _{n}]^{\times }$ .

The Kronecker–Weber theorem states that every finite abelian extension of $\mathbb {Q}$ in $\mathbb {C}$ is contained in $\mathbb {Q} (\zeta _{n})$ for some $n$ . Equivalently, the union of all the cyclotomic fields $\mathbb {Q} (\zeta _{n})$ is the maximal abelian extension $\mathbb {Q} ^{\mathrm {ab} }$ of $\mathbb {Q}$ .

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 $n\geq 3$ , the following are equivalent:

a regular $n$ -gon is constructible;
there is a sequence of fields, starting with $\mathbb {Q}$ and ending with $\mathbb {Q} (\zeta _{n})$ , such that each is a quadratic extension of the previous field;
$\varphi (n)$ is a power of 2;
$n=2^{a}p_{1}\cdots p_{r}$ for some integers $a,r\geq 0$ and Fermat primes $p_{1},\ldots ,p_{r}$ . (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 $\zeta _{3}={\tfrac {-1+{\sqrt {-3}}}{2}}$ and $\zeta _{6}={\tfrac {1+{\sqrt {-3}}}{2}}$ show that $Q (ζ 3) = Q (ζ 6) = Q (\sqrt -3)$ , which is a quadratic extension of $Q$ . Correspondingly, a regular 3-gon and a regular 6-gon are constructible.
$n = 4$ : Similarly, $ζ 4 = i$ , so $Q (ζ 4) = Q (i)$ , and a regular 4-gon is constructible.
$n = 5$ : The field $Q (ζ 5)$ is not a quadratic extension of $Q$ , but it is a quadratic extension of the quadratic extension $Q (\sqrt 5)$ , 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 $x n + y n$ , where $n$ 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 _{n}y)\ldots (x+\zeta _{n}^{n-1}y)

Here $x$ and $y$ are ordinary integers, whereas the factors are algebraic integers in the cyclotomic field $Q (ζ n)$ . If unique factorization holds in the cyclotomic integers $Z [ζ n]$ , 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 $Z [ζ n]$ has unique factorization is^[3]

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 $Z [ζ n]$ , measured the failure of unique factorization via the class number $h n$ and proved that if $h p$ 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 OEIS: A055513 or OEIS: A000927 for the $h$ -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

^ Stillwell, John (1994). Elements of Algebra. Undergraduate Texts in Mathematics. Springer New York. p. 100. doi:10.1007/978-1-4757-3976-3. ISBN 978-1-4419-2839-9.
^ Washington 1997, Proposition 2.7.
^ 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