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If K/FK/F is a finite algebraic extension, it can be defined by a polynomial f(x)F[x]f(x)\in F[x]. The polynomial discriminant, disc(f)\mathrm{disc}(f), is well-defined up a factor of a non-zero square. The discriminant root field of the extension is F(disc(f))F(\sqrt{\mathrm{disc}(f)}), which is well-defined.

If n=[K:F]n=[K:F], then the Galois group GG for K/FK/F is a subgroup of SnS_n, well-defined up to conjugation. The discriminant root field can alternatively be described as the fixed field of GAnG\cap A_n.

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  • Last edited by David Roe on 2024-04-13 10:36:47
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