This book is best understood as a daughter's tribute to her father. Nancy Albert has put together much of the public information on A. Adrian Albert's life, interviewed some of the relevant players, looked at some of Albert's correspondence, and added a few family anecdotes and personal memories. The result is interesting, if not inspired. It certainly makes the case for Albert's importance in the history of American mathematics in the 20th century. Perhaps it will motivate someone to do a fuller and more detailed biography.
The author makes much of Albert's origins as the child of lower middle-class Jewish parents who might well have gone into his parents' retail business had his father not noticed his talent and insisted that he get a college education. She then follows Albert's very quick rise from the ranks: college graduation in 1926, a master's degre in 1927, a Ph.D. in 1928 (all from the University of Chicago). Then, after brief stints at Princeton and Columbia, Albert returned to Chicago in 1931 and stayed until his retirement in 1971.
The author labors hard to set up the context for all this. We get chapters on the family's Lithuanian roots, on Jewish life in Chicago in the early twentieth century, on the two World Wars, on the Nazi takeover of Germany, and on the student unrest on various campuses in the late 1960s and early 1970s. Some of this might be necessary, but most of it is not: we get little information on how these events affected Albert (or not), and most of what is recalled is fairly common knowledge. Most of these sections should have been shortened.
Things get more troublesome when the author tackles mathematics. In order to establish context, she presents us with a potted history of algebra "From Al-Jabr to Abstract Algebra"; this is mostly embarassing. I admit that after a bit I just skipped to the end. She then tries to explain her father's contributions to algebra. She manages to convey the idea that they were impressive, but does not offer much that will help the reader understand Albert's main concerns, his mathematical style, or his most important achievements.
Perhaps the most interesting chapter is the one on Albert's role in the Brauer-Hasse-Noether theorem, which the author clearly feels should be known as the Albert-Brauer-Hasse-Noether theorem. She traces the extensive correspondence between Albert and Hasse in the early 1930s. The correspondence, which dealt with the problem of determining all central division algebras over Q and over number fields, reveals an interesting mix of collaboration and competition. She does not remark on how unusual this was: on one side, three of the leading mathematicians of Germany, clearly the world's leading center for research on algebra and number theory; on the other, competing as an equal, an American, nurtured in the old-style pre-Noether algebra, from what was still (relative to Germany) a mathematical backwater. Rather than developing this theme, the author concentrates (understandably) on trying to rescue Albert's role in the proof from the forgetfulness of the mathematical community.
The other interesting component of the book is the emphasis placed on Albert's defense-related work, starting in World War II and continuing until the end of his life. She suggests that Albert pretty much invented the idea that mathematicians could contribute to the cryptographic efforts of the American government. She also states that in more than one occasion Albert posed theoretical questions to other mathematicians that ultimately derived from his secret government work.
The family anecdotes are, surprisingly, much less interesting. Perhaps it is just that they are too closely related to the private man who is largely absent from this biography, which concentrates on his public roles. The best story, in fact, is not about Albert at all, but about his wife Frieda.
The topical arrangement of the book is the cause of some repetitiveness. The same event can be mentioned more than once, often in similar words, if it relates to two different aspects of Albert's life.
What is missing here? Well, the first thing that is missing is Albert's interior life. In a biography, one would want more about the man and what made him tick. The little we get here consists of a daughter's observations, but they rarely go beyond such things as "he was passionate about his mathematics." We might have found out more had the author looked at Albert's correspondence a bit more intensely. I get the feeling that she felt she needed to respect her parents' privacy on certain things; fair enough, but a biographer's job is to fill in the picture, going beyond the public story.
The second thing that is missing is serious engagement with Albert's mathematics. My impression of him, both before and after reading this book, is that he was something of a throwback. He came to age as a mathematician just as Emil Artin and Emmy Noether were revolutionizing algebra by the systematic introduction of the "structural" approach to the subject. My gut feeling about Albert's work is that while he could play in the "structuralist" playground as well as anyone, his fundamental approach remained more concrete, more centered on examples than on "algebraic structures." This impression, however, is not the fruit of careful study of the work itself. A biography would have been a good opportunity to assess this estimate of his position in the history of algebra in the 20th century.
Finally, I guess I must also say that what is missing is the guiding hand of an editor. This book could have been much improved with a little editorial work, some punching-up of the prose, some advice and insight from a mathematician and/or a historian of mathematics with some competence in algebra, a better index, and so on.
Still, I don't want to end on a negative note. Albert was clearly an important figure in American mathematics. He deserves more attention than he has had. His work raises interesting questions about the history of algebra in the 20th century, and also about the role of mathematics in American defense, security, and intelligence. If nothing else, Nancy Albert's tribute to her father reminds us of all this. Let's hope someone takes up the challenge.
Fernando Q. Gouvêa is professor of mathematics at Colby College in Waterville, ME. His main interests are in number theory and the history of mathematics.