# Hilbert’s tenth problem for algebraic function fields of characteristic 2

```@article{Eisentraeger2002HilbertsTP,
title={Hilbert’s tenth problem for algebraic function fields of characteristic 2},
author={Kirsten Eisentraeger},
journal={Pacific Journal of Mathematics},
year={2002},
volume={210},
pages={261-281}
}```
Let K be an algebraic function field of characteristic 2 with constant field C K . Let C be the algebraic closure of a finite field in K. Assume that C has an extension of degree 2. Assume that there are elements u, x of K with u transcendental over C K and x algebraic over C(u) and such that K = C K (u,x). Then Hilbert's Tenth Problem over K is undecidable. Together with Shlapentokh's result for odd characteristic this implies that Hilbert's Tenth Problem for any such field K of finite… Expand
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#### References

SHOWING 1-10 OF 19 REFERENCES
Hilbert’s Tenth Problem for algebraic function fields over infinite fields of constants of positive characteristic
Let K be an algebraic function field of characteristic p > 2. Let C be the algebraic closure of a finite field in K. Assume that C has an extension of degree p. Assume also that K contains a subfieldExpand
Hilbert’s tenth problem for rational function fields in characteristic 2
In Hilbert's Tenth problem for fields of rational functions over finite fields (Invent. Math. 103 (1991)) Pheidas showed that Hilbert's Tenth problem over a field of rational functions with constantExpand
Hilbert's Tenth Problem for fields of rational functions over finite fields
SummaryWe prove that there is no algorithm to solve arbitrary polynomial equations over a field of rational functions in one letter with constants in a finite field of characteristic other than 2 andExpand
Diophantine Undecidability over Algebraic Function Fields over Finite Fields of Constants
Abstract We show that Diophantine problem (otherwise known as Hilbert's Tenth Problem) is undecidable over the fields of algebraic functions over the finite fields of constants of characteristicExpand
The Diophantine Problem for Polynomial Rings of Positive Characteristic
Publisher Summary This chapter discusses the diophantine problem for polynomial rings of positive characteristic. The chapter proves that the diophantine problem is unsolvable for the ring ofExpand
The Diophantine problem for polynomial rings and fields of rational functions
We prove that the diophantine problem for a ring of polynomials over an integral domain of characteristic zero or for a field of rational functions over a formally real field is unsolvable.
Diophantine unsolvability for function fields over certain infinite fields of characteristic p
• Mathematics
• 1992
Abstract We prove that if F is a proper subfield of the algebraic closure of Z p , Diophantine equations over the function field F ( t ) are not algorithmically decidable where p is an odd prime.
Diophantine undecidability of C(t1, t2)
• Mathematics
• 1992
We give a short proof that the Diophantine theory of the complex function field C ( t 1 , t 2 ) where t i are algebraically independent over the complex numbers, in the language {+, ×, t 1 , t 2 , 0,Expand
Lectures on the theory of algebraic functions of one variable
Function fields and valuations.- The Riemann-Roch theorem.- Zeta function and L-functions.- Constant field extensions.
DIOPHANTINE EQUATIONS OVER FUNCTION FIELDS
This paper proves the boundedness of the degrees of at least two components of an arbitrary solution of the equation , where are pairwise relatively prime polynomials. Bibliography: 4 titles.