Exercise 3, § 2

(a) Let K be a field and f a place of K with values in an orderable field L (Algebra, Chapter VI, § 2, Exercise 8); show that K is orderable. (Note that, for a finite family of elements (a_i)_{1 \leq i \leq n} of K, there is always an index j such that f(a_i/a_j) \neq \infty for all i.)

If $(x_i)_{1 \leq i \leq n}$ is a sequence of elements of K such that $f\left( \sum_{i=1}^n x_i^2 \right) \neq \infty$, show that $f(x_i) \neq \infty$ for all i.

(b) Let K be an ordered field and G a subgroup of K; show that the ring F(G) of elements of K which are not infinitely great with respect to G (loc. cit., Exercise 11) is a valuation ring of K and the ideal I(G) of elements of F(G) infinitesimally small with respect to G is the maximal ideal of this ring; hence there is on K a place with values in k(G) = F(G)/I(G), which reduces to the identity on G; this is called the canonical place of K with respect to G.

(c) Show that, if there exists no extension of G contained in K, distinct from G and comparable with G, k(G) is algebraic over G (note that, if t \in K does not belong to G, the field G(t) contains an element u \neq 0 infinitesimally small with respect to G and that G(t) is algebraic over G(u)).