On the Argument of Abel, respecting the Impossibility of expressing a Root of any General Equation above the Fourth Degree, by any finite Combination of Radicals and Rational Functions.was published in the Transactions of the Royal Irish Academy, volume 18 (1839), pp. 171-259.
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An abstract of this paper was published in the Proceedings of the Royal Irish Academy.
In this paper, Hamilton gave an exposition of the method of Abel for proving the impossibility of solving the general quintic equation by radicals (giving his own arguments to cover those parts of Abel's proof which he considered problematic or obscure). Hamilton also used Abel's theory to show that there are no `irreducible' solutions of the general quadratic, cubic and biquadratic equations essentially different from the standard solutions of those equations.
An abstract of Hamilton's paper On the Argument of Abel, entitled Investigations respecting Equations of the Fifth Degree was communicated on May 22nd, 1837 and published in the Proceedings of the Royal Irish Academy, volume 1 (1841), pp. 76-80.
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