Quartic fields and radical extensions

• Authors: Huah Chu;Ming-Chang Kang
• Venue: Journal of Symbolic Computation
• Year: 2002

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 - υ ∈ 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, 81υ2 - 12u3 = w2 if char K ≠ 2, or u3/υ2 = 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- υT - w ∈ K[T] with υ ≠ 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 - υ2 = 0 and 64wX2 - (32uw - 3υ2)X + (4u2w + 16w2 - uυ2) - Y2 = 0. The situation when υ = 0 will also be solved.