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Squares. (English) Zbl 0785.11022
London Mathematical Society Lecture Note Series. 171. Cambridge: Cambridge University Press. xii, 286 p. £27.50; \$ 39.95 (1993).

The author examines problems concerning squares and sums of squares, which have, as quoted in the text, a long history. It is intended to present the results as elementary as possible and to avoid theory such as the algebraic theory of quadratic forms. In order to read the book only knowledge of basic algebra is required.

Let $K$ be a field $\left(\text{char}\phantom{\rule{4.pt}{0ex}}K\ne 2\right)$. The first main topic are identities of the form

$\left({X}_{1}^{2}+\cdots +{X}_{r}^{2}\right)\left({Y}_{1}^{2}+\cdots +{Y}_{s}^{2}\right)={Z}_{1}^{2}+\cdots +{Z}_{n}^{2}\phantom{\rule{2.em}{0ex}}\left(\mathrm{I}\right)$

where ${Z}_{l}\in K\left(\left\{{X}_{i},{Y}_{j}\right\}\right)$. Hurwitz showed that for $r=s=n$ identities of the type (I) for ${Z}_{l}$ bilinear in ${X}_{i}$, ${Y}_{j}$ exist if and only if $n=1,2,4,8$. Pfister showed that for $r=s=n$ such (nonbilinear) identities exist if and only if $n$ is a power of 2. Further the level of a field is always a 2-power and any 2-power occurs as the level of a field. The level $s\left(K\right)$ of $K$ is defined as the smallest $s\le \infty$ such that $-1$ is a sum of $s$ squares in $K$. For arbitrary $r$, $s$ the author proves theorems of Hurwitz-Radon and Gabel and examines the Hopf condition.

The second main topic are problems in connection with Hilbert’s 17th problem. The author shows that any PSD (positive semidefinite) $f\in ℝ\left({X}_{1},\cdots ,{X}_{m}\right)$ is a sum of at most ${2}^{m}$ squares, in particular the Pythagoras number of $ℝ\left({X}_{1},\cdots ,{X}_{m}\right)$ is $\le {2}^{m}$. Hilbert showed that homogeneous PSD polynomials $f\in ℝ\left[{X}_{1},\cdots ,{X}_{m}\right]$ are not necessarily sums of squares in $ℝ\left[{X}_{1},\cdots ,{X}_{m}\right]$ unless $m=2$ or $degf=2$ or $m=4$, $degf=3$. Further in this connection theorems of Calderon, Choi, Lam, Reznick and Robinson are stated.

In section 11 some basic facts on quadratic forms are introduced, then Pfister’s theory of multiplicative quadratic forms and Artin-Schreier theory of formally real fields. This is used to study the behaviour of the level and the Pythagoras number $p$ under field extension. For some special fields $K$ such as number fields $s\left(K\right)$ and $p\left(K\right)$ are determined. In particular, any PSD $f\in ℚ\left[X\right]$ is a sum of 5 squares, that is $p\left(ℚ\left(X\right)\right)=5$. To show this theorem of Pourchet, the author needs to apply the Hasse-Minkowski theorem.

Except for the application of the Hasse-Minkowski theorem the book is absolutely self-contained. It seems carefully investigated, historic remarks and exercises are added. Because of the elementary approach some proofs become very technical, but the arguments are well explained. Quoting the author the book is written in such a way that it can be easily understood by an undergraduate student. Even for the reader who is familiar with the books of Lam and Scharlau on quadratic forms this book is a good additional lecture since only few points of intersection occur.

##### MSC:
 11E25 Sums of squares, etc 12D15 Formally real fields 11-01 Textbooks (number theory) 12-01 Textbooks (field theory)