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- W2017222362 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 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 , 81 v 2 − 12 u 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 , 64 X 3 − 32 uX 2 + (4 u 2 + 16 w ) X − v 2 = 0 and 64 wX 2 − (32 uw − 3 v 2 ) X + (4 u 2 w + 16 w 2 − uv 2 ) − Y 2 = 0. The situation when v = 0 will also be solved." @default.
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- W2017222362 date "2002-07-01" @default.
- W2017222362 modified "2023-09-29" @default.
- W2017222362 title "Quartic Fields and Radical Extensions" @default.
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- W2017222362 doi "https://doi.org/10.1006/jsco.2002.0543" @default.
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