- Library Home /
- Search Collections /
- Open Collections /
- Browse Collections /
- UBC Theses and Dissertations /
- An extension to the Hermite-Joubert problem
Open Collections
UBC Theses and Dissertations
UBC Theses and Dissertations
An extension to the Hermite-Joubert problem Brassil, Matthew
Abstract
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.
Item Metadata
Title |
An extension to the Hermite-Joubert problem
|
Creator | |
Publisher |
University of British Columbia
|
Date Issued |
2016
|
Description |
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.
|
Genre | |
Type | |
Language |
eng
|
Date Available |
2016-04-27
|
Provider |
Vancouver : University of British Columbia Library
|
Rights |
Attribution-NonCommercial-NoDerivatives 4.0 International
|
DOI |
10.14288/1.0300318
|
URI | |
Degree | |
Program | |
Affiliation | |
Degree Grantor |
University of British Columbia
|
Graduation Date |
2016-05
|
Campus | |
Scholarly Level |
Graduate
|
Rights URI | |
Aggregated Source Repository |
DSpace
|
Item Media
Item Citations and Data
Rights
Attribution-NonCommercial-NoDerivatives 4.0 International