The study of algebraic number fields is arguably the backbone of all of number theory. One generally subdivides number theory into four subdisciplines, namely, elementary number theory, geometric number theory, analytic number theory, and algebraic number theory; algebraic number fields very properly belong to the latter subdiscipline. Indeed, the study of number fields is essentially coextensive with algebraic number theory, at least if one allows the inclusion of local fields into the discussion.

Analytic number theory can be characterized as largely the study of L-functions, and these are organically tied to number fields (or their ideals and maximal orders) — to be sure, the objection might be raised that analytic number theory might also be regarded as being concerned at least equally with elliptic functions, modular forms, and so on, but here we need only cite the roles of Mellin and inverse Mellin transforms to return the center of gravity to number fields.

What about geometric number theory? Well, consider that gem of a result, Minkowski’s theorem on lattice points in convex bodies: in \(\mathbb{R}^n\), if an origin-symmetric convex set has volume at least \(2^n\), then it must contain at least one integer lattice point, i.e. an element of \(\mathbb{Z}^n\). Here’s a natural generalization: if we now consider a lattice comprised of the integer points (or the maximal order — see e.g. Borevich-Shafarevich) of an algebraic number field \(k\) of degree \(n\) (and if the number field is not totally real, situate it in an appropriate \(\mathbb{C}^n\cong\mathbb{R}^{2n}\)), we get the generalization that a corresponding origin symmetric convex set is guaranteed to contain a lattice point of this flavor, if its volume is at least \(2^nD\) where \(D\) is the modulus of the determinant of any basis for the lattice. A beautiful result about, yes, algebraic number fields.

And that takes us to the somewhat notorious area of elementary number theory, loosely defined as number theory without the luxury of complex analysis. Admittedly, in this area it’s a lot harder to make connections with number fields, or at least it’s much more of a stretch. But here’s a shot: the prime number theorem, as done by Selberg and Erdős, is surely a very representative result in this connection, addressing the asymptotic behavior of the primes. Well, replace the ambient ring of integers with the maximal order of a number field and then ask about the distribution of prime ideals: same question. In fact if we require the number field’s ring of integers to be a unique factorization domain, or that the number field in question have class number 1, then we’re again talking about prime elements: emphatically the same question, generalized in a very natural way.

This said, yes, there are nonetheless a lot of arithmetical questions that avoid the study of algebraic number fields, and here, just for fun (and for the sake of balanced reporting), is one: is \(\pi+e\) irrational? But I think my point is made: algebraic number fields are everywhere, and if you think you’ve escaped them, look again, they frequently come out of the shadows very quickly. After all the integers are only a special case, a special instance of a maximal order in a number field. If, indeed, Kronecker is right, and “God gave us the integers, and all else is man’s handiwork,” then number fields are among the very first things *homo mathematicus* hit on when the world was young.

So, a book unabashedly devoted to number fields is a fabulous idea. In addition to having been re-set in proper mathematical typography, this second edition (see a review of the first edition) sports a very supportive Foreword by Barry Mazur, whose first sentence reads: “What a wonderful book this is!” Oh, and Mazur goes on: “How generous it is in its tempo, its discussion, and in the details it offers. It heads off — rather than merely clears up — the standard confusions and difficulties beginners often have…” He continues his remarks by noting that the book under review strikes a wonderful balance between text and exercises, and is indeed the book he always uses when teaching the according material. And it goes without saying that the exercises in the book — and there are many — are of great importance and the reader should certainly do a lot of them; they are very good and add to the fabulous experience of learning this material.

Thus, Marcus’ *Number Fields* is indeed a fantastic book. It is truly a terrific introduction to this subject so central to number theory, modulo the *caveat *that local fields are altogether absent from the picture. But that’s all right: these do appear on the scene later in the usual pedagogical scheme of things, and the great classics in the genre are similarly disposed. Mazur mentions the book, *Vorlesungen über die Theorie der Algebraischen Zahlen*, by Erich Hecke, which was where he (and coincidentally, I) learned the subject: no local fields. And can one possibly do algebraic number theory justice without Serre’s *Corps Locaux* as the successor to books like these? Yes, start with global fields, and Marcus is a major contender in this contest, and then go local — and I admit that even Serre has some competition: I recommend the coverage in Cassels-Fröhlich, modulo the presence of an appropriate level of mathematical maturity.

*Number Fields* travels along the right trajectory, as the chapter sequence indicates, but here are some highlights. In Chapter 3, “Prime Decomposition in Number Rings,” the author explicitly singles out the matter of the splitting of primes in extensions. This is of course how it should be, but it cannot be overstated that this is the *raison* *d’être* for these decomposition considerations, once one has the (later) bird’s eye view of class field theory (of which Marcus presents a preview at the end of the book!). Parenthetically, I do recall that in my favorite book in this area, Hecke’s aforementioned *Vorlesungen*, splitting of primes in extensions is championed as a vehicle for reciprocity laws: *Klassenkörpertheorie* is knocking loudly on the door.

But to continue with Marcus, in Chapter 4, “Galois Theory Applied to Prime Decomposition,” he devotes an entire subsection to what is sometimes kept in the shade a little too much, namely, the Frobenius automorphism. I recall that in the dark ages, when I first saw this material, it took a while for me to realize just how important this player is.

Let me mention, also, that Marcus has the very good taste to include his seventh chapter on the Dedekind zeta function and the class number formula. As the book heads to its close, the reader is presented with a gem that cannot help but tantalize.

The book is supplemented by four appendices, including one on Galois theory for subfields of the complex numbers: not only useful and on-target, but another instance of Marcus’ excellent taste.

Oh, and one more thing. Earlier we made mention of the asymptotic behavior of the primes, i.e. the beautiful fact that \(\pi(x)\sim \frac{x}{\log x}\) (the Prime Number Theorem which was proposed by Riemann in his only (!) paper on number theory, *Über die Anzahl der Primzahlen unterhalb einer Gegebenen Größe*. This paper surely qualifies as the most stunning paper ever written, given that Riemann goes on in the span of only nine pages to phrase nothing less than the Riemann Hypothesis. And this brings us to our last point about the book under review: Marcus’ last chapter is devoted to “The Distribution of Primes and an Introduction to Class Field Theory” — he is showing the reader the high country, if only from a distance.

Obviously Mazur is right: it’s a wonderful book.

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