... What makes a mathematics book great? If we were to create a prize for the best book about mathematics, what criteria would we impose against which to judge? I am not referring here to books which aim to popularise mathematics or books aimed at pre-university level, mainly because they are outside my expertise, but to books which describe some branch of mathematics that one is likely to encounter at degree level or above. To draw a little more on my historical fiction analogy, I am also less concerned here with research monographs because I want to focus more on the case where the plot is well known and the thing that distinguishes the book from the rest of the field is the quality of the presentation.

As far as I am aware there are no major existing prizes for such books. Prizes exist for improving the public understanding of mathematics (e.g., the MAA Euler Book Prize) and for research books (e.g., the AMS Joseph L. Doob Prize) but the space in between is sparsely populated and may even be empty. Pressures of modern academic life also contribute to the fact that many talented writers choose never to write a book and pass on their perspective on a branch of mathematics, preferring instead to concentrate on writing those journal articles out of which careers are made. It appears more and more that academics’ outputs need to be assessed to be valued and that this is diminishing the role of scholarship in our discipline. This is a pity. The education and/or inspiration provided to others by a quality book has immense value, and in a discipline such as mathematics good books have a long life.

What follows is a personal view of some [six, J.B.] of the books that have made a great impression on me over the last 30 years. To me, there is something about the way these books are written that makes them stand out from the crowd. The subjects covered span a very broad spectrum of mathematical topics and are pitched at varying levels of difficulty. By and large, the more advanced a topic the harder it is to write a book that really engages the reader (other than being impressed by the author’s knowledge) but there are plenty of examples to show that it can be done. I should add that many texts are incredibly useful without being much fun to read, and often are designed to be so. This is fine, my aim here is simply to identify some books which, in my view, have that extra wow factor.

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As some readers will know, I take a keen interest in the history
of mathematics and the device of using history to enliven an exposition
of mathematical ideas certainly works on me. Here I would
like to mention Bewersdorff’s *Galois Theory for Beginners: A
Historical Perspective* [5]. This is not a branch of mathematics that
I studied at undergraduate (or any other) level and my current
understanding of the main results in this area is best described as
partial. However, the book that has helped me the most in gaining
what understanding I do have is this one. The history is not in the
form of a few anecdotes about Galois’ love life or his death, but an
exposition which roughly follows the historical development of
the subject. This allows one to see various abstract notions as
natural consequences of existing previous work. I find that many
books introduce abstract constructs because generations of mathematicians
have established that these are the right settings in
which to work. The reader is expected to take this on trust and get
on with it. Maybe at some later date they will have gained enough
insight to understand why they started where they did. To some
extent, this is the nature of the beast but there are sometimes available
alternatives and Bewersdorff’s book is an excellent example.

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**CHRIS LINTON FIMA**

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[5] Bewersdorff, J. (2006) *Galois Theory for Beginners: A Historical
Perspective*, American Mathematical Society, originally published in
German in 2004.

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