Quartic Fields and Radical Extensions

Let K be a field and K(α) be an extension field ofK . If [ K(α) : K ] = 3, charK ≠ = 3, and the minimal polynomial of α over K isT3 − uT − v K [T ], it is proved in Kang (2000, Am. Math. Monthly, 107, 254–256) thatK (α) is a radical extension of K if and only if, for somew K, 81 v2 − 12u3 = w2if 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, charK ≠ = 2, and the minimal polynomial of α over K isT4 − uT2 − 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, 64 X3 − 32uX2 + (4 u2 + 16w )X − v2 = 0 and 64wX2 − (32 uw − 3v2 )X + (4 u2w + 16w2 − uv2) − Y2 = 0. The situation when v = 0 will also be solved.