Hilbert's tenth problem for algebraic function fields of characteristic 2

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Abstract

Let K be an algebraic function field of characteristic 2 with constant field CK. 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 CK and x algebraic over C(u) and such that K = CK(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 characteristic is undecidable. In particular, Hilbert's Tenth Problem for any algebraic function field with finite constant field is undecidable.

Original languageEnglish (US)
Pages (from-to)261-281
Number of pages21
JournalPacific Journal of Mathematics
Volume210
Issue number2
DOIs
StatePublished - Jun 2003

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Hilbert's Tenth Problem
Algebraic Function Fields
Transcendental
Galois field
Closure
Odd
Imply

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

Cite this

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abstract = "Let K be an algebraic function field of characteristic 2 with constant field CK. 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 CK and x algebraic over C(u) and such that K = CK(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 characteristic is undecidable. In particular, Hilbert's Tenth Problem for any algebraic function field with finite constant field is undecidable.",
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Hilbert's tenth problem for algebraic function fields of characteristic 2. / Eisenträger, Kirsten.

In: Pacific Journal of Mathematics, Vol. 210, No. 2, 06.2003, p. 261-281.

Research output: Contribution to journalArticle

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