Abstract
Is a given algebraic equation solvable by radicals? Can one solve a given algebraic equation of degree n using solutions of auxiliary algebraic equations of smaller degree and radicals? In this chapter, we discuss how Galois theory answers these questions (at least in principle).
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Notes
- 1.
Since the field K has characteristic 0, it automatically contains the two square roots of unity 1 and − 1.
- 2.
If \(z_{i}Q'(y_{i}) = T(y_{i})\) for all i, then in particular, we obtain that zQ(y) = T(y) by setting i = 1.
- 3.
The existence of such a polynomial will be established in Lemma 2.27.
- 4.
And indeed every subquotient, i.e., a quotient group of a subgroup.
References
M. Berger, Geometry, translated from French by M. Cole and S. Levy. Universitext (Springer, Berlin, 1987)
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Khovanskii, A. (2014). Solvability of Algebraic Equations by Radicals and Galois Theory. In: Topological Galois Theory. Springer Monographs in Mathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38871-2_2
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DOI: https://doi.org/10.1007/978-3-642-38871-2_2
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Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-38870-5
Online ISBN: 978-3-642-38871-2
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