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In his book Galois Theory, Ian Stewart mentions:

In $1813$ Paolo Ruffini published yet another version of his impossibility proof. The paper appeared in an obscure journal, with several gaps in the proof (Bourbaki Elements d'Histoire des Mathematiques $1969$, page $103$). The most significant omission was to assume that all radicals involved must be based on rational functions of the roots.

I do not understand the last statement of above quoted paragraph - radicals involved must be based on rational functions of the roots.

Can one explain with example, what Ruffini specially assumed in his proof, which is not correct for the proof of his assertion (of insolvability by radicals)?

Maths Rahul
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    The key idea of the proof by both Ruffini and Abel is that any radicals in the expression for root must be rational function of the roots. This ensures that all the calculations to find the root are being done in the splitting field of the polynomial. – Paramanand Singh Sep 09 '21 at 13:05
  • See for more details: https://math.stackexchange.com/a/2515874/72031 – Paramanand Singh Sep 09 '21 at 13:08

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