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 trE/F(a)=trE/F(a3)=0, that is, with minimal polynomial of the form λ7+c2λ5+c4λ3+c5λ2+c6λ+c7, where the ci'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 θ and divided by the derivative of its minimal polynomial evaluated at θ. 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 c1(a)=c3(a)=0. Moreover, every field extension of degree seven of an arbitrary field has a generator a with c1(a)=c3(a)=0.
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BibTeX
@misc{chebolu2026septic,
title = {{A septic covariant and the Hermite--Joubert problem in degree seven}},
author = {Sunil Chebolu and Ján Mináč and Behzad Nikzad and Charlotte Ure},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.15786},
eprint = {2609.15786},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.15786},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - A septic covariant and the Hermite--Joubert problem in degree seven
AU - Sunil Chebolu
AU - Ján Mináč
AU - Behzad Nikzad
AU - Charlotte Ure
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.15786
UR - https://arxiv.org/abs/2609.15786
AB - 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 tr_E/F(a)=tr_E/F(a^3)=0, that is, with minimal polynomial of the form λ^7+c_2λ^5+c_4λ^3+c_5λ^2+c_6λ+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 θ and divided by the derivative of its minimal polynomial evaluated at θ. 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.
ER -