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There are similar questions already regarding this topic but I'm hoping the answers can be diluted for me here. So regarding the fact that the roots of 5th order polynomials cannot be expressed as algebraic combinations of its coefficients a,b,c,d,e,f, in general...does this imply:

  1. That the roots live outside of the set of complex numbers that can be created from algebraic operations on the coefficients?

  2. That the roots cannot be expressed as algebraic combinations of ANY finite set of coefficients in the complex numbers (a,b,c,d,e,f, or any other number)

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  • $\begingroup$ "roots live outside of the set of complex numbers" are roots of what? Certainly there are "roots" which are not complex numbers but the corresponding equations are of another nature than you seem refered to here (for example the equation $x^2=-1$ has an non-countable infinity of roots is this $1$ is intended to be the unity of the quaternions). If you want to limit to just the ordinary polynomial equations, all the possible roots are in $\mathbb{C}$ who is algebraically closed. $\endgroup$ – Piquito Jul 15 '15 at 20:30
  • $\begingroup$ @Ataulfo he was referring to those numbers "that can be created from algebraic operations on the coefficients" $\endgroup$ – ljfa Jul 16 '15 at 13:06
  • $\begingroup$ @ljfa: as many as we want (for example $a+a+a+a+........+a$ n times. And all without radicals (supposing the coefficients rational) will stay in $\mathbb {Q}$. And when using radicals all staying in the algebraically closed field $\mathbb {C}$ $\endgroup$ – Piquito Jul 17 '15 at 13:30
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The allowed calculations are :

  • rational expressions with +,-,*,/ (for example $b+e$ , $\frac{a}{d}$ , $bc$, $4abc$). Note that rational constants may also be used.

  • n-roots (for example $(b+e)^\frac{1}{5}$)

So, a formula like $(b-\frac{a}{d})^\frac{1}{5}-(c+de)^\frac{1}{3}$ would be allowed.

Every formula which can be deducted using these two kinds of calculations would be allowed. The theorem states that no such formula solves the quintic in general.

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