Field extension of the rational numbers by a primitive root of unity
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In number theory, a cyclotomic field is a number field obtained by adjoining a complex root of unity to Q, 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 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.
theory, a cyclotomicfield is a number field obtained by adjoining a complex root of unity to Q, the field of rational numbers. Cyclotomicfields played...
example of the construction of a quadratic field is to take the unique quadratic field inside the cyclotomicfield generated by a primitive p {\displaystyle...
This geometric fact accounts for the term "cyclotomic" in such phrases as cyclotomicfield and cyclotomic polynomial; it is from the Greek roots "cyclo"...
In mathematics, the nth cyclotomic polynomial, for any positive integer n, is the unique irreducible polynomial with integer coefficients that is a divisor...
the next proof, the individual elements of the Gauss sum are in the cyclotomicfield L = Q ( ζ p ) {\displaystyle L=\mathbb {Q} (\zeta _{p})} but the above...
quadratic fields are subfields of cyclotomicfields, and implicitly deduced quadratic reciprocity from a reciprocity theorem for cyclotomicfields. His proof...
abelian group can be found as the Galois group of some subfield of a cyclotomicfield extension by the Kronecker–Weber theorem. Another useful class of examples...
commutative ring of algebraic integers in the algebraic number field Q(ω) – the third cyclotomicfield. To see that the Eisenstein integers are algebraic integers...
defined to be regular if it does not divide the class number of the pth cyclotomicfield Q(ζp), where ζp is a primitive pth root of unity. The prime number...
fields are the n-th layers of the cyclotomic Z2-extension of Q. Also in 2009, Morisawa showed that the class numbers of the layers of the cyclotomic Z3-extension...
operations in this field are defined in analogy with the case of Gaussian rational numbers, d = − 1 {\displaystyle d=-1} . The cyclotomicfield Q ( ζ n ) , {\displaystyle...
In mathematics, a cyclotomic unit (or circular unit) is a unit of an algebraic number field which is the product of numbers of the form (ζa n − 1) for...
number. This number lies in the n-th cyclotomicfield — and in fact in its real subfield, which is a totally real field and a rational vector space of dimension...
adjoining roots of unity to a field, or a subextension of such an extension. The cyclotomicfields are examples. A cyclotomic extension, under either definition...
prime, ζ is a pth root of unity and K = Q(ζ ) is the corresponding cyclotomicfield, then an integral basis of OK = Z[ζ] is given by (1, ζ, ζ 2, ..., ζ p−2)...
(OEIS: A037274) Odd primes p that divide the class number of the p-th cyclotomicfield. 37, 59, 67, 101, 103, 131, 149, 157, 233, 257, 263, 271, 283, 293...
examples include imaginary quadratic fields, cyclotomicfields, and, more generally, CM fields. Any number field that is Galois over the rationals must...
rational numbers. Cyclotomicfield An extension of the rational numbers generated by a root of unity. Totally real field A number field generated by a root...
posed in 1962 by Basil Gordon and remains unsolved. Algebraic integer Cyclotomicfield Eisenstein integer Eisenstein prime Hurwitz quaternion Proofs of Fermat's...
relationship between p-adic L-functions and ideal class groups of cyclotomicfields, proved by Kenkichi Iwasawa for primes satisfying the Kummer–Vandiver...
by studying the quadratic field F = Q ( ℓ ∗ ) {\displaystyle F=\mathbb {Q} ({\sqrt {\ell ^{*}}})} and the cyclotomicfield L = Q ( ζ ℓ ) {\displaystyle...
number field has a normal integral basis. This may be seen by using the Kronecker–Weber theorem to embed the abelian field into a cyclotomicfield. Many...
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