The text demonstrates the use of general concepts by applying theorems from various areas in the context of one problem—solving the quintic. The problem is approached from two directions: the first is Felix Klein's nineteenth-century approach, using the icosahedron. The second approach presents recent works of Peter Doyle and Curt McMullen, which update Klein's use of transcendental functions to a solution through pure iteration.

Filling a pedagogical gap in the literature and providing a solid platform from which to address more advanced material, this meticulously written book:

• Develops the Riemann sphere and its field of functions, classifies the finite groups of its automorphisms, computes for each such group a generator of the group-invariant functions, and discusses algebraic aspects of inverting this generator
• Gives, in the case of the icosahedral group, an elegant presentation of the relevant icosahedral geometry and its relation to the Brioschi quintic
• Reduces the general quintic to Brioschi form by radicals
• Proves Kronecker's theorem that an "auxiliary" square root is necessary for any such reduction
• Expounds Doyle and McMullen's development of an iterative solution to the quintic
• Provides a wealth of exercises and illustrations to clarify the geometry of the quintic