Power integral bases in the family of simplest quartic fields. (English) Zbl 1092.11042

Olajos, Péter

Exp. Math. 14, No. 2, 129-132 (2005).

Let \(K_t\) be the field generated by a root of the polynomial \(X^4-tX^3-6X^2+tX+1\) (\(t\) integral, distinct from \(0\), \(\pm3\)). The author shows that if \(t^2+16\) is not divisible by a square, then \(K_t\) does not have a power integral basis, except when \(t=2\) or \(t=4\). In these two cases all generators of power integral bases are listed.

Reviewer: Władysław Narkiewicz (Wrocław)

MSC:

11R16	Cubic and quartic extensions
11D57	Multiplicative and norm form equations
11R04	Algebraic numbers; rings of algebraic integers
11Y50	Computer solution of Diophantine equations

Keywords:

quartic fields; power integral basis

Software:

KANT/KASH