Exercise 6, § 9

Let K be a commutative field of characteristic $\neq 2$; let f be a mapping of K into K such that $f(x + y) = f(x) + f(y)$ for all x and y and for all $x \neq 0$ $f(x)f(1/x) = 1$. Show that f is an isomorphism of K onto a subfield of K (prove that $f(x^2) = (f(x))^2$).