Commentary on an unpublished lecture by G. N. Watson on solving the quintic

Birmingham, England. Some pages that had become separated f r o m the notes were found by the third author in one of the boxes during a vis i t to B i rmingham in 1999. "Solving the quintic" is one of the f ew topics in mathematics which has been of enduring and widespread interest for centuries. The history of this subject is beautifully illustrated in the poster produced by MATHEMATICA. Many attempts have been made to solve quintic equations; see, f o r example, [6]-[14], [17]-[21], [28]-[32], [34]-[36], [58]-[60]. Galois was the f i r s t mathematic ian to determ ine which quintic polynomials have roots expressible in terms of radicals, and in 1991 D u m m i t [24] gave formulae fo r the roots of such solvable quintics. A quintic is solvable by means of radicals ~f and only i f its Galois group is the cyclic group 77/5W_ of order 5, the dihedral group D5 of order 10, or the Frobenius group F2o of order 20. In view of the current interest (both theoretical and computational) in solvable quintic equations [24], [33], [43]-[46], i t seemed to the duthors to be of interest to publish Professor Watson's notes on his lecture, wi th commentary explaining some of the ideas in more current mathematical language. For those having a practical need for solving quintic equations, Watson's step-by-step procedure will be especially valuable. Watson's method applies to any solvable quintic polynomial, that is, any quintic polynomial whose Galois group is one of Z/5~, D5 or F2o. Watson's interest in solving quintics was undoubtedly motivated by his keen interest in veri fying Srinivasa Ramanujan 's determinations of class invariants, or equivalently, singular moduli. Ramanujan computed the values of over 100 class invariants, which he recorded