Annales Academiæ Scientiarum Fennicæ

Mathematica

Volumen 43, 2018, 685-692

# ON THE MINIMAL FIELD OF DEFINITION
OF RATIONAL MAPS: RATIONAL MAPS OF ODD SIGNATURE

## Rubén A. Hidalgo

Universidad de La Frontera,
Departamento de Matemática y Estadística

Temuco, Chile; ruben.hidalgo 'at' ufrontera.cl

**Abstract.**
The field of moduli of a rational map is an invariant under conjugation by
Möbius transformations.
Silverman proved that a rational map, either of even degree or equivalent to a polynomial, is definable over its field of moduli and he also provided examples of rational maps of odd degree for which such a property fails. We introduce the notion for a rational map to have odd signature and prove that this condition ensures for the field of moduli to be a field of definition. Rational maps being either of even degree or equivalent to polynomials are examples of odd signature ones.

**2010 Mathematics Subject Classification:**
Primary 37P05, 37F10, 14G05.

**Key words:**
Rational maps, field of moduli,
field of definition, Galois groups.

**Reference to this article:** R. A. Hidalgo:
On the minimal field of definition of rational maps:
rational maps of odd signature.
Ann. Acad. Sci. Fenn. Math. 43 (2018), 685-692.

Full document as PDF file

https://doi.org/10.5186/aasfm.2018.4340

Copyright © 2018 by Academia Scientiarum Fennica