### Abstract

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 characteristic is undecidable. In particular, Hilbert's Tenth Problem for any algebraic function field with finite constant field is undecidable.

Original language | English (US) |
---|---|

Pages (from-to) | 261-281 |

Number of pages | 21 |

Journal | Pacific Journal of Mathematics |

Volume | 210 |

Issue number | 2 |

DOIs | |

State | Published - Jun 2003 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

}

*Pacific Journal of Mathematics*, vol. 210, no. 2, pp. 261-281. https://doi.org/10.2140/pjm.2003.210.261

**Hilbert's tenth problem for algebraic function fields of characteristic 2.** / Eisenträger, Kirsten.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Hilbert's tenth problem for algebraic function fields of characteristic 2

AU - Eisenträger, Kirsten

PY - 2003/6

Y1 - 2003/6

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=18144437291&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=18144437291&partnerID=8YFLogxK

U2 - 10.2140/pjm.2003.210.261

DO - 10.2140/pjm.2003.210.261

M3 - Article

AN - SCOPUS:18144437291

VL - 210

SP - 261

EP - 281

JO - Pacific Journal of Mathematics

JF - Pacific Journal of Mathematics

SN - 0030-8730

IS - 2

ER -