A septic covariant and the Hermite--Joubert problem in degree seven

Abstract

We prove the degree-seven case of the Hermite--Joubert problem in characteristic zero: if $F$ is a field of characteristic zero and $E/F$ is a field extension of degree seven, then $E$ is generated by an element $a$ with ${\text{tr}}_{E/F}(a)={\text{tr}}_{E/F}(a^3)=0$, that is, with minimal polynomial of the form ${\lambda }^7+c_2{\lambda }^5+c_4{\lambda }^3+c_5{\lambda }^2+c_6\lambda +c_7$, where the $c_i$'s belong to $F$. This is given by an explicit formula: a covariant of the binary septic of coefficient degree seven and order five, evaluated at a generator $\theta$ and divided by the derivative of its minimal polynomial evaluated at $\theta$. We also announce the general theorem, which will be proved in a companion paper in preparation: over every infinite field, in every characteristic, every étale algebra of degree seven contains a primitive element $a$ with $c_1(a)=c_3(a)=0$. Moreover, every field extension of degree seven of an arbitrary field has a generator $a$ with $c_1(a)=c_3(a)=0$.

13 pages, latex 11 pt