Explicit construction of integral bases of radical function fields

Wu, Qingquan

Journal de théorie des nombres de Bordeaux, Tome 22 (2010) no. 1, pp. 259-270.

Nous donnons une construction explicite d’une base entière pour le corps de fonction $K=k(t,\rho )$, où ${\rho }^n=D\in k\left[t\right]$, sous l’hypothèse $[K:k(t\left)\right]=n$ et $char\left(k\right)\nmid n$. Le discriminant du corps $K$ est également calculé. Nous expliquons pourquoi ces questions sont considérablement plus faciles que dans le cas des corps de nombres. Quelques formules pour les $P$-signatures des corps de fonction radiciels sont également présentées dans ce papier.

We give an explicit construction of an integral basis for a radical function field $K=k(t,\rho )$, where ${\rho }^n=D\in k\left[t\right]$, under the assumptions $[K:k(t\left)\right]=n$ and $char\left(k\right)\nmid n$. The field discriminant of $K$ is also computed. We explain why these questions are substantially easier than the corresponding ones in number fields. Some formulae for the $P$-signatures of a radical function field are also discussed in this paper.

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