Cyclotomic Number Fields

1995

One of the mysteries of algebraicK-theory is its relation to classical conjectures of number theory. Before we recall some instances of the relation let us introduce the necessary notation. For an odd prime l, let F = Q(μl) and E = Q(μlk ). We fix a primitive root of unity ξlk of order l . Let A and A denote the l-Sylow subgroup of the ideal class group of F and the ith eigenspace of A under the action of the Galois group G(F/Q), respectively. Let A be the direct sum of A with i even cf. [12, p.100]. There are two famous conjectures in number theory which concern the class group of the cyclotomic field F .

Journal of Algebra, 1999

We study the automorphism groups of cyclic extensions of the rational function fields. We give conditions for the cyclic Galois group to be normal in the whole automorphism group, and then we study how the ramification type determines the structure of the whole automorphism group.

IAENG International Journal of Applied Mathematics

For each cyclic quintic field F of dis-criminant dF smaller than 2 · 10 7 , we established lists of quadratic relative extensions of absolute discrimi-nant less than 3·2 15 ·d 2 F in absolute value. For each one of the found fields, the field discriminant, the quintic field discriminant, a polynomial defining the relative quadratic extension, the corresponding relative dis-criminant, the corresponding polynomial over Q, and the Galois group of the Galois closure are given. Among the found fields, there exists for each fixed cyclic field F , 6 totally imaginary cyclic number fields and between 2 and 4 totally real cyclic number fields.

Nagoya Mathematical Journal, 1975

Let p be a prime. If one adjoins to Q all pn -th roots of unity for n = 1, 2, 3, …, then the resulting field will contain a unique subfield Q ∞ such that Q ∞ is a Galois extension of Q with Gal the additive group of p-adic integers. We will denote Gal(Q ∞/Q ) by Γ.

Journal of Pure and Applied Algebra, 1981

International Journal of Scientific and Research Publications (IJSRP), 2020

Galois theory is about the connection between groups, fields and their extensions. It is the interplay between polynomials, fields, and groups. Galois field is about fields and their extensions. It has got many useful applications in computing and some other areas of abstract algebra and it has also been used to produce many useful theorems.

Bulletin of the American Mathematical Society, 2011

Irregular primes-37 being the first such prime-have played a great role in number theory. This article discusses Ken Ribet's constructionfor all irregular primes p-of specific abelian, unramified, degree p extensions of the number fields Q(e 2πi/p). These extensions with explicit information about their Galois groups (they are Galois over Q) were predicted to exist ever since the work of Herbrand in the 1930s. Ribet's method involves a tour through the theory of modular forms; it demonstrates the usefulness of congruences between cuspforms and Eisenstein series, a fact that has inspired, and continues to inspire, much work in number theory.

Seema

A number field K is a fit field extension of Q. Its degree is [K: Q]. i.e. its dimension as Q-vector a space. An algebraic number is an algebraic integer if it satisfies a ionic polynomial with integer coefficients, equivalently. Its minimal polynomial over Q should have integer coefficients.

Journal of Number Theory, 1975

We compute the Schur group of the cyclotomic fields Q(c,,,) and real quadratic fields Q(&/*) where d is a product of an even number of primes congruent to three modulo four. Some results are also given about the Schur group of certain subfields of cyclotomic fields.