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

The Absolute Galois Group of a Rational Function Field in Characteristic Zero is a Semi-Direct Product

1984 ◽  
Vol 27 (3) ◽  
pp. 313-315 ◽  
Author(s):  
Lou Van Den Dries ◽  
Paulo Ribenboim
Keyword(s):  
Galois Group ◽  
Rational Function ◽  
Direct Product ◽  
Characteristic Zero ◽  
Function Field ◽  
Profinite Group ◽  
Absolute Galois Group ◽  
Rational Function Field ◽  
Free Profinite Group ◽  
The Absolute

AbstractLet K be a field of characteristic 0 and t an indeterminate. It is shown that the absolute Galois group of K(t) is the semi-direct product of a free profinite group with the absolute Galois group of K.

scholarly journals Indépendance algébrique de logarithmes en caractéristique P

2006 ◽  
Vol 74 (3) ◽  
pp. 461-470 ◽  
Author(s):  
Laurent Denis
Keyword(s):  
Zeta Function ◽  
Rational Function ◽  
Riemann Zeta Function ◽  
Function Field ◽  
Fundamental Period ◽  
Rational Function Field ◽  
Linearly Independent ◽  
The Riemann Zeta Function ◽  
Carlitz Module ◽  
Riemann Zeta

Let k be the rational function field over the field with q elements with characteristic p. Since the work of Carlitz we know in this situation the function ζ analog of the Riemann zeta function and the function Logφ analog of the usual logarithm. We will show two main results. Firstly, if ξ denotes the fundamental period of Carlitz module, we prove that ξ, ζ(1),…, ζ(p – 2) are algebraically independent over k. Secondly if α1,…, αn are rational elements (of degree less than q/(q − 1) to ensure convergence of the logarithm) such that Logφ α1,…, Logφ αn are linearly independent over k then they are algebraically independent over k. The point is to find suitable functions taking these values and for which Mahler's method can be used.

scholarly journals The invariants of orthogonal group actions

1993 ◽  
Vol 48 (2) ◽  
pp. 313-319 ◽  
Author(s):  
Li Chiang ◽  
Yu-Ching Hung
Keyword(s):  
Quadratic Form ◽  
Finite Field ◽  
Rational Function ◽  
Linear Group ◽  
Function Field ◽  
Orthogonal Group ◽  
Group Actions ◽  
Rational Function Field ◽  
Transcendental Extension

Let Fq be the finite field of order q, an odd number, Q a non-degenerate quadratic form on , O(n, Q) the orthogonal group defined by Q. Regard O(n, Q) as a linear group of Fq -automorphisms acting linearly on the rational function field Fq(x1, …, xn). We shall prove that the invariant subfield Fq(x1,…, xn)O(n, Q) is a purely transcendental extension over Fq for n = 5 by giving a set of generators for it.

Defining transcendentals in function fields

2002 ◽  
Vol 67 (3) ◽  
pp. 947-956 ◽  
Author(s):  
Jochen Koenigsmann
Keyword(s):  
Rational Function ◽  
Function Field ◽  
Function Fields ◽  
Algebraic Closure ◽  
Rational Function Field ◽  
First Order

AbstractGiven any field K, there is a function field F/K in one variable containing definable transcendental over K, i.e., elements in F / K first-order definable in the language of fields with parameters from K. Hence, the model-theoretic and the field-theoretic relative algebraic closure of K in F do not coincide. E.g., if K is finite, the model-theoretic algebraic closure of K in the rational function field K(t) is K(t).For the proof, diophantine ∅-definability ofK in F is established for any function field F/K in one variable, provided K is large, or K× /(K×)n is finite for some integer n > 1 coprime to char K.

scholarly journals The autocorrelation of the Möbius function and Chowla's conjecture for the rational function field in characteristic 2

2015 ◽  
Vol 373 (2040) ◽  
pp. 20140311 ◽  
Author(s):  
Dan Carmon
Keyword(s):  
Finite Field ◽  
Rational Function ◽  
Function Field ◽  
Möbius Function ◽  
Rational Function Field ◽  
Mobius Function ◽  
Characteristic 2

We prove a function field version of Chowla's conjecture on the autocorrelation of the Möbius function in the limit of a large finite field of characteristic 2, extending previous work in odd characteristic.

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