Galois Theory of Solvability
The unsolvability of general polynomial equations of degree ≥ 5 leaves us with very little to say about solvability of general equations, that is, equations with indeterminate coefficients. The general linear, quadratic, cubic and quartic equations are solvable — and that’s it. The investigation of solvability is much more fruitful in the domain of equations with numerical coefficients, where there are solvable equations of arbitrarily high degree. For this reason, all fields in this chapter are assumed to be number fields, that is, subfields of ℂ, unless there is an explicit statement to the contrary. We shall be particularly interested in number fields of finite degree over ℚ, which we know from Chapter 5 to be of the form ℚ(α1,..., α k ) where α1,..., α k are algebraic numbers.
KeywordsNormal Subgroup Galois Group Number Field Prime Order Galois Theory
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