## Kodai Mathematical Journal

- Kodai Math. J.
- Volume 34, Number 3 (2011), 410-425.

### Lifting monogenic cubic fields to monogenic sextic fields

Melisa J. Lavallee, Blair K. Spearman, and Kenneth S. Williams

#### Abstract

Let *e* {-1, +1}. Let *a,b* **Z** be such that *x*^{6} + *ax*^{4} + *bx*^{2} + *e* is irreducible in **Z**[*x*]. The cubic field *C* = **Q** (α), where α^{3} + *a*α^{2} + *b*α + *e* = 0, is said to lift to the sextic field *K* = **Q**(θ), where θ^{6} + *a*θ^{4} + *b*θ^{2} + *e* = 0. The field *K* is called the lift of *C*. If {1, α, α^{2}} is an integral basis for *C* (so that *C* is monogenic), we investigate conditions on *a* and *b* so that {1, θ, θ^{2}, θ^{3}, θ^{4}, θ^{5}} is an integral basis for the lift *K* of *C* (so that *K* is monogenic). As the sextic field *K* contains a cubic subfield (namely *C*), there are eight possibilities for the Galois group of *K*. For five of these Galois groups, we show that infinitely many monogenic sextic fields can be obtained in this way, and for the remaining three Galois groups, we show that only finitely many monogenic fields can arise in this way, when *e* {-1, +1}.

#### Article information

**Source**

Kodai Math. J., Volume 34, Number 3 (2011), 410-425.

**Dates**

First available in Project Euclid: 10 November 2011

**Permanent link to this document**

https://projecteuclid.org/euclid.kmj/1320935550

**Digital Object Identifier**

doi:10.2996/kmj/1320935550

**Mathematical Reviews number (MathSciNet)**

MR2855831

**Zentralblatt MATH identifier**

1237.11045

#### Citation

Lavallee, Melisa J.; Spearman, Blair K.; Williams, Kenneth S. Lifting monogenic cubic fields to monogenic sextic fields. Kodai Math. J. 34 (2011), no. 3, 410--425. doi:10.2996/kmj/1320935550. https://projecteuclid.org/euclid.kmj/1320935550