Whenever I read about the history of algebra, I end up with the same conclusion: They solved the general cubic, then the general quartic and then spent lots of years trying to solve the general quintic, which later was proved impossible by Abel and Galois (I guess Ruffini also made contributions to the topic).

There's a gap in time, from the first atempts at finding the general solution of the quintic to Abel's proof - what have people tried before Abel's proof? I am curious about what kind of techniques they tried to use for finding the general quintic.

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    $\begingroup$ I recommend you to read Jim Brown's paper "Abel and The Insolvability of the Quintic" for your question. math.caltech.edu/~jimlb/abel.pdf $\endgroup$ – Mathlover Oct 7 '13 at 8:12
  • $\begingroup$ @Mathlover, thanks for link to that nice paper. $\endgroup$ – lhf Oct 7 '13 at 11:50
  • $\begingroup$ For a book that specifically studies the period of time you (correctly) identified as a gap, see the book I mention in my answer to History of the theory of equations: John Colson. $\endgroup$ – Dave L. Renfro Nov 7 '13 at 13:06

I don't know much about what happened from Cardano to Lagrange, but I do know that Lagrange tried to modify the approaches that worked for the quadratic, cubic and quartic.

All those approaches can be thought of as working by constructing the quantity $x_1+\omega x_2+\omega^2 x_3+...+\omega^{n-1} x_n$ (where $x_1,..,x_n$ are the roots of the polynomial and $\omega$ is the $n$th root of unity). Constructing this quantity basically solves the problem since the roots can then be expressed from it using only the arithmetic operations. For example in the case of the quadratic if you rewrite the discriminant in terms of the roots you'll find that $D=(x_1-x_2)^2$ and $\sqrt{D}=x_1-x_2$ which is the above expression for $n=2$.

In the case of the quintic, let $\zeta$ be the primitive $5$th root of unity and let $x_1,x_2,...,x_5$ be the roots of the polynomial you are trying to solve.

We're trying to express the quantity $x_1+\zeta x_2+\zeta^2 x_3+\zeta^3 x_4+\zeta^4 x_5$.

One way (and the way that was used for all lower degree equations) one can try to find this quantity is by solving the equation $\prod_{\sigma \in S_n} (X-\sigma(x_1+\zeta x_2+\zeta^2 x_3+\zeta^3 x_4+\zeta^4 x_5))=0$ - in other words a polynomial in $X$ formed taking the product of terms $(X-\sigma(x_1+\zeta x_2+\zeta^2 x_3+\zeta^3 x_4+\zeta^4 x_5))$ where $\sigma$ runs over all permutations of $x_1,...,x_5$.

This polynomial has coefficients which are symmetric in $x_1,...,x_5$ and can hence be expressed as rational functions of the coefficients of the original equation.

While this may look like a $120$ degree polynomial it is actually a $24$ degree polynomial in $X^5$. And here the approach stops.

A similar approach will reduce a quadratic to a linear equation, a cubic to a quadratic and a quartic to a sextic which can still be solved due to some of its special properties. But Lagrange wasn't able to make this approach work for the quintic.


You may enjoy reading these books:

  • $\begingroup$ These books appear to only be tangentially related to the question. Or perhaps they actually address attempts made? If so, can you give a short summary? $\endgroup$ – RghtHndSd Oct 7 '13 at 11:38
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    $\begingroup$ @rghthndsd, Bewersdorff discusses early attempts and special cases in chapter 8. $\endgroup$ – lhf Oct 7 '13 at 11:42

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