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Quartic Fields and Radical Extensions

✍ Scribed by Huah Chu; Ming-chang Kang

Publisher: Elsevier Science
Year: 2002
Tongue: English
Weight: 224 KB
Volume: 34
Category: Article
ISSN: 0747-7171
DOI: 10.1006/jsco.2002.0543

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✦ Synopsis

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 Kang (2000, Am. Math. Monthly, 107, 254-256)

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 2 w + 16w 2 -uv 2 ) -Y 2 = 0. The situation when v = 0 will also be solved.

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