Cyclotomic field

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In number theory, a cyclotomic field is a number field obtained by adjoining a complex primitive root of unity to Q, the field of rational numbers. The n-th cyclotomic field Qn) (where n > 2) is obtained by adjoining a primitive n-th root of unity ζn to the rational numbers.

The 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.

Properties[edit]

A cyclotomic field is the splitting field of the cyclotomic polynomial

xn − 1

and therefore it is a Galois extension of the field of rational numbers. The degree of the extension

[Qn):Q]

is given by φ(n) where φ is Euler's phi function. A complete set of Galois conjugates is given by { (ζn)a } , where a runs over the set of invertible residues modulo n (so that a is relative prime to n). The Galois group is naturally isomorphic to the multiplicative group

(Z/nZ)×

of invertible residues modulo n, and it acts on the primitive nth roots of unity by the formula

b: (ζn)a → (ζn)a b.

Relation with regular polygons[edit]

Gauss made early inroads in the theory of cyclotomic fields, in connection with the geometrical problem of constructing a regular n-gon with a compass and straightedge. His surprising result that had escaped his predecessors was that a regular heptadecagon (with 17 sides) could be so constructed. More generally, if p is a prime number, then a regular p-gon can be constructed if and only if p is a Fermat prime; in other words if \varphi(p)=2^k is a power of 2.

For n = 3 and n = 6 primitive roots of unity admit a simple expression via square root of three, namely:

ζ3 = 3i − 1/2,   ζ6 = 3i + 1/2

Hence, both corresponding cyclotomic fields are identical to the quadratic field Q(−3). In the case of ζ4 = i = −1 the identity of Q4) to a quadratic field is even more obvious. This is not the case for n = 5 though, because expressing roots of unity requires square roots of quadratic integers, that means that roots belong to a second iteration of quadratic extension. The geometric problem for a general n can be reduced to the following question in Galois theory: can the nth cyclotomic field be built as a sequence of quadratic extensions?

Relation with Fermat's Last Theorem[edit]

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

xn + yn = zn

as follows:

xn + yn = (x + y) (x + ζy) … (x + ζn − 1y).

Here x and y are ordinary integers, whereas the factors are algebraic integers in the cyclotomic field Qn). If unique factorization of algebraic integers were true, then it could have been 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. Unfortunately, the unique factorization fails in general – for example, for n = 23 – but Kummer found a way around this difficulty. He introduced a replacement for the prime numbers in the cyclotomic field Qp), expressed the failure of unique factorization quantitatively via the class number hp and proved that if hp is not divisible by p (such numbers 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 using it, established Fermat's theorem for all prime exponents p less than 100, with the exception of 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 to Cyclotomic Field[edit]

(sequence A061653 in OEIS), or OEISA055513 (for prime n)

n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Class number 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
n 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40
Class number 1 1 3 1 1 1 1 1 8 1 9 1 1 1 1 1 37 1 2 1
n 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Class number 121 1 211 1 1 3 695 1 43 1 5 3 4889 1 10 2 9 8 41241 1
n 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80
Class number 76301 9 7 17 64 1 853513 8 69 1 3882809 3 11957417 37 11 19 1280 2 100146415 5
n 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100
Class number 2593 121 838216959 1 6205 211 1536 55 13379363737 1 53872 201 6795 695 107692 9 411322824001 43 2883 55
n 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120
Class number 3547404378125 5 9069094643165 351 13 4889 63434933542623 19 161784800122409 10 480852 468 1612072001362952 9 44697909 10752 132678 41241 1238459625 4
n 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140
Class number 12188792628211 76301 8425472 75456 57708445601 7 2604529186263992195 359057 37821539 64 28496379729272136525 11 157577452812 853513 75961 111744 646901570175200968153 69 1753848916484925681747 39
n 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160
Class number 1257700495 3882809 36027143124175 507 1467250393088 11957417 5874617 4827501 687887859687174720123201 11 2333546653547742584439257 1666737 2416282880 1280 84473643916800 156 56234327700401832767069245 100146415 223233182255 31365

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