Английская Википедия:Cyclotomic field

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Шаблон:Short description Шаблон:More footnotes needed In number theory, a cyclotomic field is a number field obtained by adjoining a complex root of unity to Шаблон:Math, 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 Шаблон:Mvar) – 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 Шаблон:Math, let Шаблон:Math; this is a primitive Шаблон:Mvarth root of unity. Then the Шаблон:Mvarth cyclotomic field is the extension Шаблон:Math of Шаблон:Math generated by Шаблон:Math.

Properties

• The Шаблон:Mvarth cyclotomic polynomial

<math>

\Phi_n(x) = \prod_\stackrel{1\le k\le n}{\gcd(k,n)=1}\!\!\! \left(x-e^{2\pi i k/n}\right) = \prod_\stackrel{1\le k\le n}{\gcd(k,n)=1}\!\!\! (x-{\zeta_n}^k) </math>

is irreducible, so it is the minimal polynomial of Шаблон:Math over Шаблон:Math.

• The conjugates of Шаблон:Math in Шаблон:Math are therefore the other primitive Шаблон:Mvarth roots of unity: Шаблон:Math for Шаблон:Math with Шаблон:Math.
• The degree of Шаблон:Math is therefore Шаблон:Math, where Шаблон:Mvar is Euler's totient function.
• The roots of Шаблон:Math are the powers of Шаблон:Math, so Шаблон:Math is the splitting field of Шаблон:Math (or of Шаблон:Math) over Шаблон:Math.
• Therefore Шаблон:Math is a Galois extension of Шаблон:Math.
• The Galois group <math>\operatorname{Gal}(\mathbf{Q}(\zeta_n)/\mathbf{Q})</math> is naturally isomorphic to the multiplicative group <math>(\mathbf{Z}/n\mathbf{Z})^\times</math>, which consists of the invertible residues modulo Шаблон:Mvar, which are the residues Шаблон:Math with Шаблон:Math and Шаблон:Math. The isomorphism sends each <math>\sigma \in \operatorname{Gal}(\mathbf{Q}(\zeta_n)/\mathbf{Q})</math> to Шаблон:Math, where Шаблон:Mvar is an integer such that Шаблон:Math.
• The ring of integers of Шаблон:Math is Шаблон:Math.
• For Шаблон:Math, the discriminant of the extension Шаблон:Math isШаблон:Sfn

<math>(-1)^{\varphi(n)/2}\,

\frac{n^{\varphi(n)}} {\displaystyle\prod_{p|n} p^{\varphi(n)/(p-1)}}.</math>

• In particular, Шаблон:Math is unramified above every prime not dividing Шаблон:Mvar.
• If Шаблон:Mvar is a power of a prime Шаблон:Mvar, then Шаблон:Math is totally ramified above Шаблон:Mvar.
• If Шаблон:Mvar is a prime not dividing Шаблон:Mvar, then the Frobenius element <math>\operatorname{Frob}_q \in \operatorname{Gal}(\mathbf{Q}(\zeta_n)/\mathbf{Q})</math> corresponds to the residue of Шаблон:Math in <math>(\mathbf{Z}/n\mathbf{Z})^\times</math>.
• The group of roots of unity in Шаблон:Math has order Шаблон:Mvar or Шаблон:Math, according to whether Шаблон:Mvar is even or odd.
• The unit group Шаблон:Math is a finitely generated abelian group of rank Шаблон:Math, for any Шаблон:Math, by the Dirichlet unit theorem. In particular, Шаблон:Math is finite only for Шаблон:Math}. The torsion subgroup of Шаблон:Math is the group of roots of unity in Шаблон:Math, which was described in the previous item. Cyclotomic units form an explicit finite-index subgroup of Шаблон:Math.
• The Kronecker–Weber theorem states that every finite abelian extension of Шаблон:Math in Шаблон:Math is contained in Шаблон:Math for some Шаблон:Mvar. Equivalently, the union of all the cyclotomic fields Шаблон:Math is the maximal abelian extension Шаблон:Math of Шаблон:Math.

Relation with regular polygons

Gauss made early inroads in the theory of cyclotomic fields, in connection with the problem of constructing a [[regular polygon|regular Шаблон:Mvar-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 Шаблон:Math, the following are equivalent:

• a regular Шаблон:Mvar-gon is constructible;
• there is a sequence of fields, starting with Шаблон:Math and ending with Шаблон:Math, such that each is a quadratic extension of the previous field;
• Шаблон:Math is a power of 2;
• <math>n=2^a p_1 \cdots p_r</math> for some integers Шаблон:Math and Fermat primes <math>p_1,\ldots,p_r</math>. (A Fermat prime is an odd prime Шаблон:Mvar such that Шаблон:Math 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

• Шаблон:Math and Шаблон:Math: The equations <math>\zeta_3 = \tfrac{-1+\sqrt{-3}}{2}</math> and <math>\zeta_6 = \tfrac{1+\sqrt{-3}}{2}</math> show that Шаблон:Math, which is a quadratic extension of Шаблон:Math. Correspondingly, a regular 3-gon and a regular 6-gon are constructible.
• Шаблон:Math: Similarly, Шаблон:Math, so Шаблон:Math, and a regular 4-gon is constructible.
• Шаблон:Math: The field Шаблон:Math is not a quadratic extension of Шаблон:Math, but it is a quadratic extension of the quadratic extension Шаблон:Math, 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 Шаблон:Math, where Шаблон:Mvar is an odd prime, appearing in one side of Fermat's equation

<math>x^n + y^n = z^n</math>

as follows:

<math>x^n + y^n = (x + y)(x + \zeta y)\cdots (x + \zeta^{n-1} y)</math>

Here Шаблон:Mvar and Шаблон:Mvar are ordinary integers, whereas the factors are algebraic integers in the cyclotomic field Шаблон:Math. If unique factorization holds in the cyclotomic integers Шаблон:Math, 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 Шаблон:Math and Euler's proof for Шаблон:Math can be recast in these terms. The complete list of Шаблон:Mvar for which Шаблон:Math has unique factorization isШаблон:Sfn

• 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 Шаблон:Math, measured the failure of unique factorization via the class number Шаблон:Math and proved that if Шаблон:Math is not divisible by a prime Шаблон:Mvar (such Шаблон:Mvar are called regular primes) then Fermat's theorem is true for the exponent Шаблон:Math. Furthermore, he gave a criterion to determine which primes are regular, and established Fermat's theorem for all prime exponents Шаблон:Mvar 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

Шаблон:OEIS, or Шаблон:Oeis or Шаблон:Oeis for the <math>h</math>-part (for prime n)

See also

• Kronecker–Weber theorem
• Cyclotomic polynomial

References

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
• Шаблон:Citation
• 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. Шаблон:ISBN

Further reading

• Шаблон:Cite book
• Шаблон:Mathworld
• Шаблон:Springer
• On the Ring of Integers of Real Cyclotomic Fields. Koji Yamagata and Masakazu Yamagishi: Proc, Japan Academy, 92. Ser a (2016)