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UBC Theses and Dissertations

An extension to the Hermite-Joubert problem Brassil, Matthew


Let E/F be a field extension of degree n. A classical problem is to find a generating element in E whose characteristic polynomial over F is as simple is possible. An 1861 theorem of Ch. Hermite [5] asserts that for every separable field E/F of degree n there exists an element a ∈ E whose characteristic polynomial is of the form f(x) = x⁵ +b₂x³ +b₄x+b₅ or equivalently, tr{E/F}(a) = tr{E/F}(a³) = 0. A similar result for extensions of degree 6 was proven by P. Joubert in 1867; see [6]. In this thesis we ask if these results can be extended to field extensions of larger degree. Specifically, we give a necessary and sufficient condition for a field F, a prime p and an integer n ≥ 3 to have the following property: Every separable field extension E/F of degree n contains an element a ∈ E such that a generates E over F, and tr{E/F}(a) = tr{E/F}(a^p) = 0. As a corollary we show for infinitely many new values of n that the theorems of Hermite and Joubert do not extend to field extensions of degree n. We conjecture the same for more values of n and provide computational evidence for a large number of these.

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