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 primitiveth root of unity. Then the th cyclotomic field is the field extension of generated by .
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 , 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;
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, ζ4 = i, 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 n for which has unique factorization is[3]
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 primes37, 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 -adic zeta functions.
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