Digging deep to the roots of the power

The Equation that Couldn't Be Solved
September 15, 2006

In the late 1820s, the work of two extraordinary mathematicians, the Norwegian Niels Henrik Abel and the Frenchman Evariste Galois, changed the course of algebra and ushered in a new era of modern mathematics. Ending a centuries-old quest, Abel proved that it is impossible to find solutions of the general quintic equation (one in which the highest power of the unknown is 5) that can be expressed in terms of the coefficients using only the four elementary algebraic operations and the extraction of roots. However, this proof did not explain how to determine whether any given quintic was solvable (in this sense) or not: the equation x5=32 is plainly solvable by x=5V 32=2, but what about x5 - 4x + 2 =0? Galois went much further. He analysed the general problem of solution by radicals, and gave a criterion for solvability that included unsolvability in the general case as one among many corollaries. To do this, he invented a whole new branch of mathematics - group theory - and showed how it could be used in the theory of equations.

Mario Livio's book is a lively addition to the small number of popular accounts of these climactic developments in mathematics. His tale is many-stranded. There is, first, the poignant drama of the lives of these two young geniuses, one of whom died in poverty at the age of 26, the other as a result of a duel when only 20, neither having enjoyed recognition.

Their stories are ably detailed in two central chapters, together comprising a quarter of the book.

Next there is the fascinating history of the millennia-long search for solutions to algebraic equations, which Livio covers entertainingly in another chapter. We learn how the ancient Babylonians, the Greeks and the Hindu mathematicians of the 7th century could all solve quadratic equations of various kinds; and how a book written in Baghdad in the 9th century summarised the theory of equations for many succeeding centuries - and gave us our word "algebra".

In the first half of the 16th century, a colourful and competitive bunch of Italians became world champions at solving cubics, and then quartics. Hopes were high that the quintic would soon yield: after all, why should 5 be that different from 4? Yet nearly three centuries passed while some of the most brilliant mathematicians tried in vain to find algebraic solutions to the quintic.

Abel's unsolvability proof ended that fruitless search, but it was Galois who found the key to unlock the deep secrets of the quintic. That key was symmetry, as formalised in the powerful conceptual structure of group theory, which he created. The third and largest strand in Livio's book is the story of symmetry itself, which he introduces in the first two chapters with many striking examples from art, music and the sciences. A more technical discussion of the development of group theory is given in chapter six, including its application to geometry by Felix Klein. The central role of symmetry and group theory in much of 20th-century physics is outlined in chapter seven, and some psychological reflections are offered in chapter eight.

In these areas, Livio faces substantial competition from several recent popular books on symmetry and group theory. But his is the only one, I think, that centres the story in the historical context of the theory of equations. And this leads us to the remaining and, potentially, the most interesting strand of all: how did group theory answer the solvability problem?

Explaining this at popular level is a severe challenge. Galois's theory is presented in mathematics departments at the advanced undergraduate/postgraduate level as one of the pinnacles of a course in modern algebra. Livio gives a commendable three-page summary of the ingredients of Galois's unsolvability proof in chapter six. I had hoped for more given the title of the book.

Indeed, more information lurks in later pages. Discussing Klein's programme, Livio emphasises the isomorphism between the order-60 group A5 of even permutations of five objects and the symmetry group of icosahedron, but omits to mention that the latter - and hence A5 - has no proper normal subgroup. Sixty pages later we learn that this was precisely the property that Galois used to prove unsolvability. These remarks, suitably amplified, could have been usefully integrated into the earlier discussion of composition factors, for this is the reason why one of the composition factors of S5 (the maximum symmetry group of the quintic) is the non-prime number 60. It is just here that the difference between 5 and 4 or 3 emerges: the composition factors for the analogous groups S4 and S3 (the latter analysed by Livio) are primes, precisely as required for solvability of quartics and cubics.

Ian Aitchison is emeritus professor of physics, Oxford University.

The Equation that Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry

Author - Mario Livio
Publisher - Souvenir Press
Pages - 353
Price - £20.00
ISBN - 0 285 63743 6

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