Quartic fields and radical extensions
Journal
Journal of Symbolic Computation
Journal Volume
34
Journal Issue
1
Pages
83-89
Date Issued
2002
Author(s)
Abstract
Let K be a field and K (α) be an extension field of K. If [K(α) : K] = 3, char K ≠ 3, and the minimal polynomial of α over K is T3 - uT - v ε K[T], it is proved in Kang (2000, Am. Math. Monthly, 107, 254-256) that K (α) is a radical extension of K if and only if, for some w ε K, 81v2 -12u3 = w2 if char K ≠ 2, or u3/v2 = w2 + w if char K = 2. In this paper, we prove a similar result when [K(α) : K] = 4, char K ≠ 2, and the minimal polynomial of α over K is T4 - uT2 - vT - w ε K[T] with v ≠ 0 : K(α) is a radical extension of K if and only if the following system of polynomial equations is solvable in K, 64X3 - 32uX2 + (4u2 + 16w)X - v2 = 0 and 64wX2 - (32uw - 3v2)X + (4u2w + 16w2 - uv2) - Y2 = 0. The situation when v = 0 will also be solved. © 2002 Elsevier Science Ltd. All rights reserved.
Type
journal article