Posting has slowed a little bit this month because of holidays, but in the last couple weeks during my visit home I decided to refresh some basic knowledge of valuation theory by going through thoroughly the book “Introduction to *p*-adic Numbers and Valuation Theory” by George Bachman. Naturally, I wrote this quick review.

Bachman’s book is designed to be a leisurely introduction to valuation theory and *p*-adic numbers. It has only 152 pages and naturally cannot be comprehensive. It is, rather, an enjoyable read that does not require much advanced knowledge, though some experience with metric spaces is certainly required to fully appreciate the later chapters on the extension of valuations.

The first two chapters provide the definition of a valuation of rank 1 (i.e. where the ordered group is a subgroup of the real numbers), and give a glimpse into the world of $ p$-adic numbers. Basic results about these are proved, and enough detail on the arithmetic of $ \mathbb{Q}_p$ is provided so that the reader can then do the basic arithmetical operations without difficulty. The basic properties of the exponential, logarithmic, and binomial functions are also developed. The theorems Newton’s method giving criteria for the existence of roots of certain polynomials are proved. This in turn is useful in the extension problem, which is developed in later chapters, which is to determine the number of extensions of a valuation to an appropriate extension field.

The $ p$-adic fields are not studied in much more detail beyond this. In particular, $ p$-adic integers and the topology of $ \mathbb{Q}_p$ are not discussed, although the reader will certainly be able to tackle these topics in more advanced texts and papers after reading this short introduction.

The remaining chapters discuss the extension problem and more general valuations, although the main focus of the book is certainly rank 1 valuations. Chapter 3 proves a general extension theorem of places, which is tedious but is the machinery behind much of the rest of the book. The language of places is also used cleverly to prove that the set of integral elements with respect to a ring map constitute a ring. This result is usually proved more prosaically by considering module conditions for integral closure.

The extension of valuations itself is developed using the theory of Banach spaces. All of the results needed are proved, and none of them are very difficult. In particular, Gel’fand’s theorem that a commutative Banach algebra over $ \mathbb{C}$ that is a field is itself isomorphic to a subfield of $ \mathbb{C}$ is shown.

For rank $ 1$ valuations, the extension problem for finite extensions is solved in detail, and a few results proved on the existence for arbitrary extensions. In more detail, given a field $ k$ and a (rank 1!) valuation on $ k$ together with a finite extension $ K$, the bijection between distinct extensions of the valuation to $ K$ and nonconjugate embeddings of $ K$ into the algebraic closure of the completion of $ k$ is done in detail. For separable extensions, this gives a correspondence further to distinct irreducible factors (in the completion!) of the minimal polynomial of $ \alpha$ where $ K=k(\alpha)$. Further refinements in the case of embedding $ K$ into a minimal $ E/k$ Galois are given. Finally, some basic results similar to the $ p$-adic setting are proved for the discrete valuation setting.

I felt the last chapter detailing the extensions of valuations could have been more motivated, perhaps with a few more comments on the relation of valuations with algebraic number theory. Unfortunately, only the first two chapters have exercises. With a book this leisurely, I feel that exercises in all chapters, as well as additional exercises in the first two chapters, would have made this book much more educational, especially since none of the existing exercises are very difficult.

The book is well written and enjoyable, taking me about ten days of reading for about an hour or two, and despite the minor issues I mentioned above, it provides enough intuition for the subject so that the interested reader will have little trouble in continuing in the subject. I should also mention that it was also helpful in refreshing some of the basic facts on $ p$-adic numbers that I forgot. Here are a few recommendations for further reading:

- Cassels and Frohlich (ed), “Algebraic Number Theory”: for those that have a good background in algebra and that are interested in number theory
- Mahler, “
*p*-adic Numbers and Their Functions”: more details on*p*-adic numbers and analysis, and more exercises