Resolvable by radicals

Algebra Level 5

The formula \[\frac{ - b \pm \sqrt{b^2 - 4ac}}{2a}\] is well known to obtain the solutions of the general equation of second degree \(ax^2 + bx + c = 0 \). When it is possible to express the solutions of a polynomial equation by using only operations with its coefficients: sum, substraction, product, division and extraction of roots (\(\sqrt{ \cdot }, \space \sqrt[3]{\cdot}, ...\)) it is said that the equation is "resolvable by radicals."

The general equation of degree n, \(p(x) = a_n x^n + a_{n -1}x^{n -1} + ... + a_1 x + a_0 = 0\) is "resolvable by radicals" if and only if \(n \leq a\). What is \(a\)?

Details.-

1) \(a\) is the maximum value for the general equation of degree n to be resolvable by radicals

2) \(p(x) \in \mathbb{R}[x], \space a_n \neq 0\)

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