field extension Definition and 1 Threads

My solution: 1. ##K(a^2) \subseteq K(a)##. 2. ##a## is zero of the quadratic polynomial ##X^2 - (a^2)##, i.e., ##[K(a) : K(a^2)] \leq 2##. 3. It is not 2 because ##[K(a) : K] = [K(a) : K(a^2)][K(a^2) : K]## is odd. 4. Thus, it is 1, and hence ##K(a) = K(a^2)##. Is this solid enough?