Construction of fields of algebraic numbers with given solvable Galois group

1956 ◽  
pp. 185-237 ◽  
Author(s):  
I. R. Šafarevič
Keyword(s):  
Galois Group ◽  
Algebraic Numbers

scholarly journals Mahler measure on Galois extensions

2018 ◽  
Vol 14 (06) ◽  
pp. 1605-1617 ◽  
Author(s):  
Francesco Amoroso
Keyword(s):  
Symmetric Group ◽  
Galois Group ◽  
Galois Extension ◽  
Mahler Measure ◽  
Algebraic Numbers ◽  
Galois Extensions

We study the Mahler measure of generators of a Galois extension with Galois group the full symmetric group. We prove that two classical constructions of generators give always algebraic numbers of big height. These results answer a question of Smyth and provide some evidence to a conjecture which asserts that the height of such a generator grows to infinity with the degree of the extension.

Decidable regularly closed fields of algebraic numbers

1985 ◽  
Vol 50 (2) ◽  
pp. 468-475 ◽  
Author(s):  
Lou van den Dries ◽  
Rick L. Smith
Keyword(s):  
Galois Group ◽  
Elementary Theory ◽  
Algebraic Closure ◽  
Closed Field ◽  
Algebraic Numbers ◽  
Integer Polynomials ◽  
Forthcoming Book ◽  
Closed Fields ◽  
Free Profinite Group ◽  
Theoretic Argument

A field K is regularly closed if every absolutely irreducible affine variety defined over K has K-rational points. This notion was first isolated by Ax [A] in his work on the elementary theory of finite fields. Later Jarden [J2] and Jarden and Kiehne [JK] extended this in different directions. One of the primary results in this area is that the elementary properties of a regularly closed field K with a free Galois group (on either finitely or countably many generators) are determined by the set of integer polynomials in one indeterminate with a zero in K. The method of proof employed in [J1], [J2] and [JK] is unusual for algebra since it is a measure-theoretic argument. In this brief summary we have not made any attempt at completeness. We refer the reader to the recent paper of Cherlin, van den Dries, and Macintyre [CDM] and to the forthcoming book by Fried and Jarden [FJ] for a more thorough discussion of the latest results. We would like to thank Moshe Jarden, Angus Macintyre, and Zoe Chatzidakis for their comments on an earlier version of this paper.A countable field K is ω-free if the absolute Galois group , where is the algebraic closure of K and is the free profinite group on ℵ0 generators.

Effective Galois theory

1981 ◽  
Vol 46 (2) ◽  
pp. 385-392 ◽  
Author(s):  
Peter La Roche
Keyword(s):  
Galois Group ◽  
Galois Theory ◽  
Galois Groups ◽  
Profinite Group ◽  
Classical Literature ◽  
Profinite Groups ◽  
Algebraic Numbers ◽  
Recursively Enumerable ◽  
Free Profinite Group ◽  
Infinite Degree

Krull [4] extended Galois theory to arbitrary normal extensions, in which the Galois groups are precisely the profinite groups (i.e. totally disconnected, compact, Hausdorff groups). Metakides and Nerode [7] produced two recursively presented algebraic extensionsK⊂Fof the rationals such thatFis abelian,Fis of infinite degree overK, and the Galois group ofFoverK, although of cardinalityc, has only one recursive element (viz. the identity). This indicated the limits of effectiveness for Krull's theory. (The Galois theory offiniteextensions is completely effective.) Nerode suggested developing a natural effective version of Krull's theory (done here in §1).It is evident from the classical literature that the free profinite group on denumerably many generators can be obtained effectively as the Galois group of a recursive extension of the rationals over a subfield. Nerode conjectured that it could be obtained effectively as the Galois group of the algebraic numbers over a suitable subfield (done here in §2). The case of finitely many generators was done non-effectively by Jarden [3]. The author believes that the denumerable case, as presented in §2, is also new classically. Using this result and the effective Krull theory, every “co-recursively enumerable” profinite group is effectively the Galois group of a recursively enumerable field of algebraic numbers over a recursive subfield.

scholarly journals ON THE ℤp-RANKS OF TAMELY RAMIFIED IWASAWA MODULES

2013 ◽  
Vol 09 (06) ◽  
pp. 1491-1503 ◽  
Author(s):  
TSUYOSHI ITOH ◽  
YASUSHI MIZUSAWA ◽  
MANABU OZAKI
Keyword(s):  
Galois Group ◽  
Number Field ◽  
Rational Number ◽  
Quadratic Field ◽  
Linear Independence ◽  
Prime Numbers ◽  
Imaginary Quadratic Field ◽  
Algebraic Numbers ◽  
Finite Set ◽  
Explicit Formulae

For a finite set S of prime numbers, we consider the S-ramified Iwasawa module which is the Galois group of the maximal abelian pro-p-extension unramified outside S over the cyclotomic ℤp-extension of a number field k. In the case where S does not contain p and k is the rational number field or an imaginary quadratic field, we give the explicit formulae of the ℤp-ranks of the S-ramified Iwasawa modules by using Brumer's p-adic version of Baker's theorem on the linear independence of logarithms of algebraic numbers.

ARITHMETIC TRACES OF NON-HOLOMORPHIC MODULAR INVARIANTS

2010 ◽  
Vol 06 (01) ◽  
pp. 69-87 ◽  
Author(s):  
ALISON MILLER ◽  
AARON PIXTON
Keyword(s):  
Modular Form ◽  
Modular Forms ◽  
Fourier Coefficients ◽  
Algebraic Numbers ◽  
Heegner Points ◽  
Positive Weight ◽  
Poincare Series ◽  
Half Integral Weight ◽  
Integral Weight ◽  
Modular Invariants

We extend results of Bringmann and Ono that relate certain generalized traces of Maass–Poincaré series to Fourier coefficients of modular forms of half-integral weight. By specializing to cases in which these traces are usual traces of algebraic numbers, we generalize results of Zagier describing arithmetic traces associated to modular forms. We define correspondences [Formula: see text] and [Formula: see text]. We show that if f is a modular form of non-positive weight 2 - 2 λ and odd level N, holomorphic away from the cusp at infinity, then the traces of values at Heegner points of a certain iterated non-holomorphic derivative of f are equal to Fourier coefficients of the half-integral weight modular forms [Formula: see text].

scholarly journals The differential Galois group of the rational function field

2021 ◽  
Vol 381 ◽  
pp. 107605
Author(s):  
Annette Bachmayr ◽  
David Harbater ◽  
Julia Hartmann ◽  
Michael Wibmer
Keyword(s):  
Galois Group ◽  
Rational Function ◽  
Function Field ◽  
Differential Galois Group ◽  
Rational Function Field

scholarly journals The Galois group of Xp+aXp−1+a

2009 ◽  
Vol 129 (4) ◽  
pp. 824-830 ◽  
Author(s):  
B. Bensebaa ◽  
A. Movahhedi ◽  
A. Salinier
Keyword(s):  
Galois Group

Pointwise bound for ℓ-torsion in class groups: Elementary abelian extensions

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Jiuya Wang
Keyword(s):  
Galois Group ◽  
Prime Number ◽  
Finite Groups ◽  
Upper Bound ◽  
Abelian Groups ◽  
Class Groups ◽  
Abelian Extensions ◽  
Pointwise Bound ◽  
First Case

AbstractElementary abelian groups are finite groups in the form of {A=(\mathbb{Z}/p\mathbb{Z})^{r}} for a prime number p. For every integer {\ell>1} and {r>1}, we prove a non-trivial upper bound on the {\ell}-torsion in class groups of every A-extension. Our results are pointwise and unconditional. This establishes the first case where for some Galois group G, the {\ell}-torsion in class groups are bounded non-trivially for every G-extension and every integer {\ell>1}. When r is large enough, the unconditional pointwise bound we obtain also breaks the previously best known bound shown by Ellenberg and Venkatesh under GRH.

Sign in / Sign up

Export Citation Format

Share Document