Quartic Fields and Radical Extensions

摘要

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 T~3 -uT - v ∈ 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 ∈K, 81v~2 - 12u~3 = w~2 if char K = 2, or u~3/v~2 = w~2 + 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 T~4 - uT~2 - vT - w ∈ 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, 64X~3 - 32uX~2 + (4u~2 + 16w)X - v~2 = 0 and 64wX~2 - (32uw - 3v~2)X + (4u~2w + 16w~2 - uv~2) - Y~2 = 0. The situation when v = 0 will also be solved.