scholarly journals Galois theory, motives and transcendental numbers

2009 ◽  
pp. 165-177 ◽  
Author(s):  
Yves André
Keyword(s):  
Galois Theory ◽  
Transcendental Numbers

scholarly journals The Completed Finite Period Map and Galois Theory of Supercongruences

2018 ◽  
Vol 2019 (23) ◽  
pp. 7379-7405
Author(s):  
Julian Rosen
Keyword(s):  
Algebraic Number ◽  
Complex Number ◽  
Galois Theory ◽  
Rational Numbers ◽  
Multiple Zeta Values ◽  
Period Map ◽  
Transcendental Numbers ◽  
Multiple Zeta ◽  
Zeta Values ◽  
Finite Period

Abstract A period is a complex number arising as the integral of a rational function with algebraic number coefficients over a region cut out by finitely many inequalities between polynomials with rational coefficients. Although periods are typically transcendental numbers, there is a conjectural Galois theory of periods coming from the theory of motives. This paper formalizes an analogy between a class of periods called multiple zeta values and congruences for rational numbers modulo prime powers (called supercongruences). We construct an analog of the motivic period map in the setting of supercongruences and use it to define a Galois theory of supercongruences. We describe an algorithm using our period map to find and prove supercongruences, and we provide software implementing the algorithm.

Field Extensions and Galois Theory

1984 ◽  
Author(s):  
Julio R. Bastida ◽  
Roger Lyndon
Keyword(s):  
Galois Theory ◽  
Field Extensions

Galois theory of equations

1972 ◽  
pp. 243-266
Author(s):  
R. Kochendörffer
Keyword(s):  
Galois Theory

ON EXCEPTIONAL SETS: THE SOLUTION OF A PROBLEM POSED BY K. MAHLER

2016 ◽  
Vol 94 (1) ◽  
pp. 15-19 ◽  
Author(s):  
DIEGO MARQUES ◽  
JOSIMAR RAMIREZ
Keyword(s):  
Entire Functions ◽  
Complex Conjugation ◽  
Exceptional Set ◽  
Exceptional Sets ◽  
Transcendental Numbers ◽  
Lecture Notes ◽  
Transcendental Entire Functions

In this paper, we shall prove that any subset of $\overline{\mathbb{Q}}$, which is closed under complex conjugation, is the exceptional set of uncountably many transcendental entire functions with rational coefficients. This solves an old question proposed by Mahler [Lectures on Transcendental Numbers, Lecture Notes in Mathematics, 546 (Springer, Berlin, 1976)].

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