Solvability by radicals of a polynomial
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7/7/2026, 5:08:47 PM · Sequoia
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Recorded titleSolvability by radicals of a polynomial
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A [[Polynomial|polynomial]] $P\in K[X]$ is solvable by radicals if...1
A polynomial equation $P(x)=0, P\in \mathbb{K}[X]$ is solvable by radicals if all its solutions can be expressed only using additions, substractions, products, divisions and $n^{th}$-roots of elements from $\mathbb{K}$.2
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For instance, $x^{2}+bx+c=0$ is solvable by radicals for all $b,c\in\mathbb{K}$ since the solutions take the form $\displaystyle\frac{-b\pm\sqrt{b^{2}-4c}}{2}$.4
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In fact, any polynomial equation of degree $3$ (Cartan's formulas) and $4$ (Ferrari's formulas) are solvable by radicals. However it does not exist any general formulas for the fifth degre, this is Abel's theorem. 6
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The theory that states if a polynomial is solvable by radicals or not is Galois theory.Revision 291
7/7/2026, 12:56:32 PM · Ancient Tree
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A [[Polynomial|polynomial]] $P\in K[X]$ is solvable by radicals if...