Exercise 5, § 9

Let K be a commutative field of characteristic $\neq 2$; let G be a subgroup of the additive group of K such that, if H denotes the set consisting of 0 and the inverses of the elements $\neq 0$ of G, H is also a subgroup of the additive group of K. Show that there exists an element $a \in K$ and a subfield $K'$ of K such that $G = aK'$ (establish first that, if x and y are elements of G such that $y \neq 0$, then $x^2/y \in G$; deduce that, if $x, y, z$ are elements of G such that $z \neq 0$, then $xy/z \in G$).