Geometry of the Quintic

This book helps students at the advanced undergraduate and beginning graduate levels to develop connections between the algebra, geometry, and analysis that they know, and to better appreciate the totality of what they have learned.

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