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Two families of monogenic $S_4$ quartic number fields

Volume 186 / 2018

Hanson Smith Acta Arithmetica 186 (2018), 257-271 MSC: 11R04, 11R09 11R16. DOI: 10.4064/aa180423-24-8 Published online: 12 October 2018


Consider the integral polynomials $f_{a,b}(x)=x^4+ax+b$ and $g_{c,d}(x)=x^4+cx^3+d$. Suppose $f_{a,b}(x)$ and $g_{c,d}(x)$ are irreducible, $b\,|\, a$, and the integers $$ b,\quad d,\quad 256d-27c^4,\quad\hbox{and}\quad \dfrac{256b^3-27a^4}{\operatorname{gcd}(256b^3,27a^4)} $$ are all square-free. Using the Montes algorithm, we show that a root of $f_{a,b}(x)$ or $g_{c,d}(x)$ defines a monogenic extension of $\mathbb{Q}$ and serves as a generator for a power basis of the ring of integers. In fact, we show monogeneity for slightly more general families. Further, we obtain lower bounds on the density of polynomials generating monogenic $S_4$ fields within the families $f_{b,b}(x)$ and $g_{1,d}(x)$.


  • Hanson SmithDepartment of Mathematics
    University of Colorado
    Campus Box 395
    Boulder, CO 80309-0395, U.S.A.

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