Exercise 29, § 2

Let K be a maximal ordered field and let G be a subfield of K. Show that the set of extensions of G contained in K, and which are comparable to G, is inductively ordered ; if E, is a maximal element of this set, show that $E_0$ is isomorphic to the field $K(G)$ defined in Ex. 11, b) of VI, p. 40 (prove that the natural map from $F(G)$ onto $K(G)$ maps E, onto $K(G)$, by first showing that $E_0$ is a maximal ordered field, using Ex. 14, a) of VI, p. 40, and then that no element of $K(G)$ is transcendent! over the image of $E_0$, using Ex. 24).