Suppose I have a field extension K of F with basis $\{1,\beta\}$, $\beta\in K^*/F^*$.
How do I show that $\beta^2$ cannot be written as $c_1+c_2\beta$, where $c_1,c_2\in F, c_2\ne 0$ unless $\beta^2 \in F^*$?
For example, if $K=\mathbb{Q}(\sqrt 2)$, $F=\mathbb{Q}$, I would want to show that $2$ cannot be written as $c_1 +c_2 \sqrt{2}$, where $c_i \in \mathbb{Q}, c_2\ne 0$, unless 2 is in $\mathbb{Q^*}$, which it is. This follows by arguing that $\sqrt{2}$ is irrational, but how do I do it in the more general setting as in above?
Sincere thanks for help.