Transcendental and Algebraic Numbers

In my youth, now long ago, I had the good fortune to be at UCLA when the number theory faculty there included the late Ernst Straus. After being exposed, in my sophomore year, to his fabulous, if demanding, lectures in the regular number theory sequence I was hooked: I ended up taking on the order of half a dozen more courses with him, not just in number theory, but also in complex analysis and graduate algebra — a fabulous experience all the way round. So it came to pass that late in my stay there, I attended his seminar on transcendental number theory, run together with the late Basil Gordon and the late David Cantor, the latter having done his doctorate under the two former. (It is striking, that at this stage, these three professors whom I knew and to whom I owe so much, are now deceased: tempus fugit, and may God rest their souls).

The seminar was fabulous, with two of the texts in the game being C. L. Siegel’s Transcendental Numbers and Alan Baker’s modern classic, Transcendental Number Theory: books with their own individual charm and austerity. Siegel was, well, Siegel: huge amounts of very clever and dense analysis, leading to very powerful results indeed; Baker, on the other hand, with his famous focus on linear forms in logarithms, started his book with an explosion of ultimately very useful notation and preliminary theorems (also heavily laden with nasty analysis), all in order to smooth over what came later, i.e. the work he won a Fields Medal for. In either case, one crawls through the prose line by line, but it’s all worth the effort.

I also recall a few marvelous “named theorems” from those days, e.g., the Gelfond-Schneider Theorem, to the effect that an algebraic irrational power of an algebraic base is transcendental. I had first heard of this result, in the form of the striking particular statement that \(2^{\sqrt{2}}\) is a transcendental number, in Constance Reid’s book, Hilbert, where she went on to discuss the notorious Göttingen practice of nostrification: evidently Gelfond had shown that \(2^{i\sqrt{2}}\) was transcendental, after which Siegel showed that indeed \(2^{\sqrt{2}}\) was transcendental. Hilbert apparently went on to give all the credit to Siegel, evincing the attitude that everything of mathematical significance took place at Göttingen — nostrification par excellence.

In any event, there is no doubt that the first result, by Gelfond, took care of Hilbert’s 7th Problem (from the Paris list), and has great historical significance if only for that reason. Therefore the book under review, Gelfond’s Transcendental and Algebraic Numbers, is of commensurate historical interest. The section of the book dealing with the aforementioned result is §2. “The Euler-Hilbert Problem,” of Chapter III, “Arithmetic properties of the set of values of an analytic function whose argument assumes values in an algebraic [number] field; transcendence properties.” To wit, on p. 106: “Theorem II: The number \(a^b\) is irrational transcendental when \(a\neq0,1\) and \(b\) are algebraic.” Without doubt a stunning result, but the fact that this famous theorem has to wait till about halfway through the book to make its appearance illustrates the intrinsic difficulty of this entire field.

Thus, although there is ostensibly nothing in the book that requires more than a solid grasp of the basic properties of algebraic numbers (the notion of algebraic independence, in particular), old-style analysis (inequalities, estimates, and so on), linear algebra (with zest), and differential equations, it is characteristic of the subject that the various maneuvers in the proofs, particularly the various arcane estimates, are of exquisite intricacy. So, reading this book, or better studying it, requires solid commitment, a great deal of mathematical maturity, and a willingness to doodle all over the margins (or elsewhere: actually, a lot of paper is called for …). Yes, the book is very dense, but, as I already said in regard to two other fine books on transcendental number theory, it’s all worth it. Indeed, just to add another bit of tantalization, cf. p. 38: “Inequality (110) enables us to give a new proof of the fact that the number of [algebraic number] fields with class number one is finite.” Wow!

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.