Hamilton published a paper entitled
On Equations of the Fifth Degree: and especially on a certain System of Expressions connected with those Equations, which Professor Badano has lately proposed.which appeared in the Transactions of the Royal Irish Academy, volume 19 (1842), pp. 329-376.
This paper is available in the following formats:
An abstract of this paper was published in the Proceedings of the Royal Irish Academy.
In this paper, Hamilton gave refuted the claim of Gerolamo Badano to have produced a solution of the general quintic polynomial equation by radicals, by demonstrating (in more than one way) that expressions that Badano had claimed would be symmetric functions of the roots of such an equation (and therefore expressible in terms of its coefficients) did not in fact have the required symmetry properties.
An abstract of Hamilton's paper On Equations of the Fifth Degree, entitled On a Method proposed by Professor Badano for the Solution of Algebraic Equations was communicated on August 4th, 1842 and published in the Proceedings of the Royal Irish Academy, volume 2 (1844), pp. 275-276.
This abstract is available in the following formats: