On a Theorem of Hermite and Joubert

Reichstein, Zinovy

Canadian journal of mathematics, Tome 51 (1999) no. 1, pp. 69-95

Résumé

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 $\text{UD}\left( n \right)$ .

Détail

DOI : 10.4153/CJM-1999-005-x

Mots-clés : 12E05, 16K20

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