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The fundamental algebra theorem asserts that every polynomial $P(z)$ of degree $n \ge 1$ (i.e. not constant), on a algebraically closed field (like the field of complex numbers), of the type:

$$P(z)=a_nz^n+ \cdots +a_1z + a_0. \quad a_i\in\Bbb C $$

admits exactly $n$ roots in that field. From this theorem it follows that a complex polynomial admits exactly $n$ complex roots (counted with the right multiplicity), while a real polynomial admits at most $n$ real roots.

If $n=1,2$ there are the formulas for the resolution. For $n=3$ the Cardano's formulas and if $n\geq 4$ problems begin except for symmetric, biquadratic, monomial equations or special cases. Why is it that even today there are no closed formulas and we always have to use of the approximate solutions? Is there something new that I do not know?

Supposing that I have an equation of $10$ th degree:

$$a_{9}x^{10}+\cdots +a_{0}=0, \quad a_i\in\Bbb C$$

Is it, actually, possible to find all the $10$ solutions?

PS: I ask something that is different from the question given as duplicate because it is for students of an high school and I ask something of very simple.

Sebastiano
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  • Starting from $n=5$, things don't necessarily have a closed formula in the general case. See Abel-Ruffini theorem. – Simon Marynissen Aug 08 '20 at 18:58
  • @SimonMarynissen Surely I have forgotten the Ruffini theorem. – Sebastiano Aug 08 '20 at 19:03
  • Even if there were a general formula for $n\geq 5$ like we have for $n\leq 4$ it would be so complicated that it would be useless in practice (compared to a numerical method). Even for $n=4$ where a formula do exist, hardly anybody uses it to solve a polynomial. – Winther Aug 08 '20 at 21:18
  • @Winther I was wondering how someone hasn't been searching for the general formula for the solutions. – Sebastiano Aug 08 '20 at 21:40
  • They have, but the work of Abel put a stop to this as there are no general formula (in radicals) though one can find formulas for special classes of polynomials (but often in terms of very complicated functions so very unlike the simple formulas we know for $n\leq 4$). – Winther Aug 08 '20 at 21:58
  • It's a bit like looking for a fraction $p/q$ that equals $\sqrt2$. We know you won't find one. – Empy2 Aug 11 '20 at 12:45
  • Galois theory explains this sort of thing, but isn't taught until third or fourth year undergraduate. – Empy2 Aug 11 '20 at 12:51
  • @Empy2 Thank you very much: for this reason I have asked the reopen of my question. – Sebastiano Aug 11 '20 at 15:26

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