## A note on defining transcendentals in function fields.(English)Zbl 1202.03045

In this note the authors fix some arguments in [J. Koenigsmann, “Defining transcendentals in function fields”, J. Symb. Log. 67, No. 3, 947–956 (2002; Zbl 1015.03041)] about function fields of one variable, say $$F/K$$. More precisely, they fix an argument in the proof of Theorem 1 on the (first-order) definability of $$K$$ in $$F$$ in case some group of $$n$$-th powers $${K^\times}^n$$ has finite index in $$K^\times$$ with $$n>1$$ coprime with the characteristic of $$K$$. Unfortunately, the new definition obtained is no longer existential. They also fix an argument in the proof of Theorem 3 on the existence for any $$K$$ of a function field of one variable $$F/K$$ which has definable transcendentals in case $$K$$ is not perfect. The authors give some further remarks of interest, e.g. that it follows from simple known facts and Koenigsmann’s key basic characterization, that every inseparable function field of one variable has definable transcendentals.

### MSC:

 03C60 Model-theoretic algebra 12F10 Separable extensions, Galois theory 12L12 Model theory of fields

Zbl 1015.03041
Full Text:

### References:

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