## Abstract

A well-known cancellation problem of Zariski asks when, for two given domains (fields) K_{1} and K_{2} over a field k, a k-isomorphism of K_{1} [t] (K_{1} (t)) and K_{2} [t] (K_{2} (t)) implies a k-isomorphism of K_{1} and K_{2}. The main results of this article give affirmative answer to the two low-dimensional cases of this problem:. 1. Let K be an affine field over an algebraically closed field k of any characteristic. SupposeK (t) ≃ k (t_{1}, t_{2}, t_{3}), thenK ≃ k (t_{1}, t_{2}) . 2. Let M be a 3-dimensional affine algebraic variety over an algebraically closed field k of any characteristic. LetA = K [x, y, z, w] / M be the coordinate ring of M. SupposeA [t] ≃ k [x_{1}, x_{2}, x_{3}, x_{4}], thenfrac (A) ≃ k (x_{1}, x_{2}, x_{3}), wherefrac (A) is the field of fractions of A. In the case of zero characteristic these results were obtained by Kang in [Ming-chang Kang, A note on the birational cancellation problem, J. Pure Appl. Algebra 77 (1992) 141-154; Ming-chang Kang, The cancellation problem, J. Pure Appl. Algebra 47 (1987) 165-171]. However, the case of finite characteristic is first settled in this article, that answered the questions proposed by Kang in [Ming-chang Kang, A note on the birational cancellation problem, J. Pure Appl. Algebra 77 (1992) 141-154; Ming-chang Kang, The cancellation problem, J. Pure Appl. Algebra 47 (1987) 165-171].

Original language | English |
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Pages (from-to) | 2235-2242 |

Number of pages | 8 |

Journal | Journal of Algebra |

Volume | 319 |

Issue number | 6 |

DOIs | |

State | Published - 15 Mar 2008 |

Externally published | Yes |

## Keywords

- Birational cancellation problems
- Cancellation conjecture of Zariski
- Good embeddings
- Lüroth's theorem