Exercise 28, § 2

a) Let K be a subfield of R and let 0 be a real number, algebraic over K. Show that the number of structures on $\mathbf{K}(\theta)$ of an ordered extension of K is equal to the number of real conjugates of 0 (use Ex. 14, a) (VI, p. 40) and 26, a)). b) Let K be a subfield of R, equipped with $n$ distinct structures of an ordered extension of Q, all archimedean ; let $P_i$ ($1 \leq i \leq n$) be the sets of positive elements under these orderings. Show that there exists a family of $n$ elements $b_i$ of K ($1 \leq i \leq n$) such that $b_i \in P_i$ for all i and $b_i \notin P_k$ for all $k \neq i$. (For each pair of distinct indices $i, k$, let $a_{ik} \in P_k$ be such that $a_{ik} \notin P_i$; consider the elements

where r is a sufficiently large integer.) c) Let $K = \mathbf{Q}(\theta)$ be an algebraic extension of Q contained in R, and let $n$ be the number of real conjugates of $\theta$. Let C be the set of sums of squares of elements of K ; show that $C^* = C \cap K^*$ is a subgroup of $K^*$ and that $(K^*:C^*) = 2^n$. (Use b) and Cor. 2 of VI, p. 23.) \*