A Worked out Galois Group for the Classroom
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Date
2024
Publication Type
Journal Article
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Abstract
Let f=X⁶ - 3X² - 1 ∈ Q[X] and let Lf be the splitting field of f over Q. We show by hand that the Galois group Gal(Lf/Q) of the Galois extension Lf/Q is isomorphic to the alternating group A4. Moreover, we show that the six roots of f correspond to the six edges of a tetrahedron and that the four roots of the polynomial X⁴ + 18X² − 72X + 81 correspond to the four faces of a tetrahedron, which allows us to determine all eight proper intermediate fields of the extension Lf/Q.
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published
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131 (6)
Pages / Article No.
501 - 510
Publisher
Taylor & Francis
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03874 - Hungerbühler, Norbert / Hungerbühler, Norbert
08848 - Halbeisen, Lorenz (Tit.-Prof.) / Halbeisen, Lorenz (Tit.-Prof.)