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Do you know how I could compute the inverse function of the following polynomial?

$f(x) = x^5+x^3+x$

Thanks in advance.

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Wolfram|Alpha only gives numerical approximate solutions for $f(x)=2$, so there is probably no formula for the inverse of $f$, though I haven't checked that the Galois group of $f$ is not solvable. –  lhf Dec 1 '12 at 19:59

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up vote 4 down vote accepted

This function probably won't have a nice inverse, since finding the inverse is equivalent to solving the equation $f(x)=c$, or equivalently finding the roots of the equation $x^5+x^3+x-c=0$. This is a quintic polynomial, and as such probably will likely not have a general solution (i.e. for all $c$) in terms of radicals. The best you can do is usually some kind of numerical method.

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It is not true that all quintics are unsolvable by radicals. This one might be. –  lhf Dec 1 '12 at 20:05
    
that's what I see, thanks. –  Rakisbro Dec 1 '12 at 20:09
    
Well, you would have to show every polynomial in the family $x^5+x^3+x-c$ is solvable. This sounds more difficult than showing a single polynomial isn't solvable (I'm not a Galois theory expert though) –  icurays1 Dec 1 '12 at 20:10
    
Yes. But my point is that the Abel–Ruffini theorem applies only to the general quintic, and the one in question is not general. –  lhf Dec 1 '12 at 20:12
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We're not looking for the roots of $f(x)$, we're looking for the inverse of $f$, so roots of $f(x)-c$. –  icurays1 Dec 1 '12 at 20:22

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