## Kyoto Journal of Mathematics

### Very good and very bad field generators

#### Abstract

Let $\mathbf {k}$ be a field. A field generator is a polynomial $F\in\mathbf {k}[X,Y]$ satisfying $\mathbf {k}(F,G)=\mathbf {k}(X,Y)$ for some $G\in\mathbf {k}(X,Y)$. If $G$ can be chosen in $\mathbf {k}[X,Y]$, we call $F$ a good field generator; otherwise, $F$ is a bad field generator. These notions were first studied by Abhyankar, Jan, and Russell in the 1970s. The present paper introduces and studies the notions of “very good” and “very bad” field generators. We give theoretical results as well as new examples of bad and very bad field generators.

#### Article information

Source
Kyoto J. Math., Volume 55, Number 1 (2015), 187-218.

Dates
First available in Project Euclid: 13 March 2015

https://projecteuclid.org/euclid.kjm/1426252135

Digital Object Identifier
doi:10.1215/21562261-2848160

Mathematical Reviews number (MathSciNet)
MR3323532

Zentralblatt MATH identifier
1321.14044

#### Citation

Cassou-Noguès, Pierrette; Daigle, Daniel. Very good and very bad field generators. Kyoto J. Math. 55 (2015), no. 1, 187--218. doi:10.1215/21562261-2848160. https://projecteuclid.org/euclid.kjm/1426252135

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