A generalization of Schm\"udgen's Positivstellensatz is given which holds for any basic closed semialgebraic set in $\mathbb{R}^n$ (compact or not). The proof is an extension of W\"ormann's proof.

A preordering $T$ is constructed in the polynomial ring $A = \R [t_1,t_2, \dots]$ (countably many variables) with the following two properties: (1)~~For each $f \in A$ there exists an integer $N$ such that $-N \le f(P) \le N$ holds for all $P \in \Sper_T(A)$. (2)~~For all $f \in A$, if $N+f, N-f \in T$ for some integer $N$, then $f \in \R$. This is in sharp contrast with the Schm\"udgen-W\"ormann result that for any preordering $T$ in a finitely generated $\R$-algebra $A$, if property~(1) holds, then for any $f \in A$, $f > 0$ on $\Sper_T(A) \Rightarrow f \in T$. Also, adjoining to $A$ the square roots of the generators of $T$ yields a larger ring $C$ with these same two properties but with $\Sigma C^2$ (the set of sums of squares) as the preordering.

Resum\'e. On caract\'erise les corps $K$ satisfaisant le th\'eor\`eme de l'axe principal \`a l'aide de propri\'et\'es des formes carac\-t\'erisation de ces m\^emes corps due \`a Waterhouse, on retrouve \`a partir de l\`a, de fa\c{c}on \'el\'ementaire, un r\'esultat de Becker selon lequel un pro-$2$-groupe qui se r\'ealise comme groupe de Galois absolu d'un tel corps $K$ est engendr\'e par des involutions. ABSTRACT. We characterize general fields $K$, satisfying the Principal Axis Theorem, by means of properties of trace forms of the finite extensions of $K$. From this and Waterhouse's characterization of the same fields, we rediscover, in quite an elementary way, a result of Becker according to which a pro-$2$-group which occurs as the absolute Galois group of such a field $K$, is generated by