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Cyclotomic field - Wikipedia

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Field extension of the rational numbers by a primitive root of unity

In algebraic number theory, a cyclotomic field is a number field obtained by adjoining a root of unity to Q {\displaystyle \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 {\displaystyle 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.

For n ≥ 1 {\displaystyle n\geq 1} , let

ζ n = e 2 π i / n ∈ C {\displaystyle \zeta _{n}=e^{2\pi i/n}\in \mathbb {C} } .

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

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

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

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 {\displaystyle n\geq 3} , the following are equivalent:

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

A natural approach to proving Fermat's Last Theorem is to factor the binomial x n + y n {\displaystyle x^{n}+y^{n}} , where n {\displaystyle n} is an odd prime, appearing in one side of Fermat's equation

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

as follows:

x n + y n = ( x + y ) ( x + ζ n y ) … ( x + ζ n n − 1 y ) {\displaystyle x^{n}+y^{n}=(x+y)(x+\zeta _{n}y)\ldots (x+\zeta _{n}^{n-1}y)}

Here x {\displaystyle x} and y {\displaystyle y} are ordinary integers, whereas the factors are algebraic integers in the cyclotomic field Q ( ζ n ) {\displaystyle \mathbb {Q} (\zeta _{n})} . If unique factorization holds in the cyclotomic integers Z [ ζ n ] {\displaystyle \mathbb {Z} [\zeta _{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 {\displaystyle n=4} and Euler's proof for n = 3 {\displaystyle n=3} can be recast in these terms. The complete list of n for which Z [ ζ n ] {\displaystyle \mathbb {Z} [\zeta _{n}]} 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 Z [ ζ n ] {\displaystyle \mathbb {Z} [\zeta _{n}]} , measured the failure of unique factorization via the class number h n {\displaystyle h_{n}} and proved that if h p {\displaystyle h_{p}} is not divisible by a prime p {\displaystyle p} (such p {\displaystyle p} are called regular primes) then Fermat's theorem is true for the exponent n = p {\displaystyle n=p} . Furthermore, he gave a criterion to determine which primes are regular, and established Fermat's theorem for all prime exponents p {\displaystyle 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 {\displaystyle p} -adic zeta functions.

(sequence A061653 in the OEIS), or OEIS: A055513 or OEIS: A000927 for the h {\displaystyle h} -part (for prime n)

• Kronecker–Weber theorem
• Cyclotomic polynomial

• 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

• Coates, John; Sujatha, R. (2006). Cyclotomic Fields and Zeta Values. Springer Monographs in Mathematics. Springer-Verlag. ISBN 3-540-33068-2. Zbl 1100.11002.
• Weisstein, Eric W. "Cyclotomic Field". MathWorld.
• "Cyclotomic field", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Yamagata, Koji; Yamagishi, Masakazu (2016). "On the ring of integers of real cyclotomic fields". Proc. Japan Academy, Series A, Math. Sciences. 92 (6). doi:10.3792/pjaa.92.73. ISSN 0386-2194.

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