Hamilton published the following papers on the solvability of polynomial equations, and in particular on the question as to whether or not the general quintic polynomial is or is not solvable by radicals:
`Theorems respecting Algebraic Elimination, connected with the Question of the Possibility of resolving in Finite Terms the General Equation of the Fifth Degree'
`Inquiry into the Validity of a Method recently proposed by George B. Jerrard, Esq. for Transforming and Resolving Equations of Elevated Degrees' (1836)
`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' (1837).and
`On Equations of the Fifth Degree: and especially on a certain System of Expressions connected with those Equations, which Professor Badano has lately proposed' (1843).
Following the solution of general cubic and biquadratic (quartic) equations by Italian mathematicians in the sixteenth century, mathematicians had sought to find solve polynomial equations of the fifth and higher degrees, but without success. Ruffini claimed, and attempted to prove, that a general solution of a quintic polynomial `by radicals' (i.e., using only the operations of addition, subtraction, multiplication, division and extraction of nth roots) was impossible. A proof of this result was published by Abel in 1826. Galois had also made a penetrating study of the solvability of polynomial equations, but his work was only brought to the attention of the mathematical community by Joseph Liouville in the 1840s.
Abel's proof was not however universally acknowledged to be complete. In particular, George Peacock, in his `Report on the Recent Progress and Present State of Certain Branches of Analysis', presented at the third meeting of the British Association for the Advancement of Science in 1833, had stated that ``some parts of it are obscure, and not perfectly conclusive''. Some mathematicians continued to present what they claimed to be solutions of the general quintic equation.
George B. Jerrard presented a paper at the meeting of the British Association for the Advancement of Science in Dublin in 1835, in which he claimed to have found a general solution to the quintic equation. Hamilton was asked to report on this, which he did the following day, expressing reservations about the purported solution. Hamilton subsequently prepared a substantial report, for the next meeting of the British Association, in Bristol in 1836, entitled `Inquiry into the Validity of a Method recently proposed by George B. Jerrard, Esq. for Transforming and Resolving Equations of Elevated Degrees', in which he showed that, although Jerrard had not in fact completely solved the general quintic equation by radicals, he had reduced the problem to one of finding the roots of a polynomial of the form
z5 + z - c = 0where the coefficient c is some complex number. (This reduction had in fact been accomplished by the Swedish mathematician E. S. Bring in 1786.)
Hamilton studied in detail Abel's proof of the impossibility of solving the general quintic equation by radicals. His exposition of Abel's proof, `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 laid before the Royal Irish Academy on May 22nd 1837, and was published in the Transactions of the Royal Irish Academy. A brief account of Hamilton's investigations was published in the Proceedings of the Royal Irish Academy.
Hamilton also investigated attempts at a general solution of the quintic equation by the Rev. R. Murphy, Fellow of Caius College, Cambridge, and by Girolamo Badano, Professor of Mathematics in the University of Genoa. In each case he found flaws in the purported solutions: certain expressions, claimed to be symmetric functions of the roots of the polynomial, were shown by Hamilton to be unsymmetric.