Integral bases and monogenity of the simplest sextic fields
Tom 183 / 2018
Acta Arithmetica 183 (2018), 173-183
MSC: Primary 11R04; Secondary 11Y50.
DOI: 10.4064/aa170502-23-10
Opublikowany online: 22 March 2018
Streszczenie
Let be an integer, m\neq -8,-3,0,5 such that m^2+3m+9 is square free. Let \alpha be a root of f=x^6-2mx^5-(5m+15)x^4-20x^3+5mx^2+(2m+6)x+1. The totally real cyclic fields K=\mathbb Q(\alpha) are called simplest sextic fields and are well known in the literature.
Using a completely new approach we find an explicit integral basis of K in parametric form and we show that the structure of this integral basis is periodic in m with period length 36. We prove that K is not monogenic except for a few values of m, in which cases we give all generators of power integral bases.
