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On a Theorem of Hermite and Joubert

Published online by Cambridge University Press:  20 November 2018

Zinovy Reichstein
Zinovy Reichstein*
Affiliation:
Department of Mathematics, Oregon State University, Corvallis, OR 97331-4605, USA email: zinovy@math.orst.edu
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Abstract

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A classical theorem of Hermite and Joubert asserts that any field extension of degree $n\,=\,5\,\text{or}\,\text{6}$ is generated by an element whose minimal polynomial is of the form ${{\lambda }^{n}}\,+\,{{c}_{1}}{{\lambda }^{n-1}}\,+\,\cdot \cdot \cdot +\,{{c}_{n-1}}\lambda \,+\,{{c}_{n}}$ with ${{c}_{1\,}}\,=\,\,{{c}_{3}}\,=\,0$. We show that this theorem fails for $n\,=\,{{3}^{m}}$ or ${{3}^{m}}+{{3}^{l}}$ (and more generally, for $n={{p}^{m}}$ or ${{p}^{m}}+{{p}^{l}}$, if 3 is replaced by another prime $p$), where $m\,>\,1\,\ge \,0$. We also prove a similar result for division algebras and use it to study the structure of the universal division algebra $\text{UD}\left( n \right)$.

We also prove a similar result for division algebras and use it to study the structure of the universal division algebra $\text{UD}\left( n \right)$.

Keywords

12E0516K20
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 51 , Issue 1 , 01 February 1999 , pp. 69 - 95
Copyright
Copyright © Canadian Mathematical Society 1999

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