Linear equations in field extension

Question

Let $F\subset E$ be a finite field extension of degree $2$ and $\,$ $^{-}:E\rightarrow E$ the conjugacy map. Consider the equation of the form $$ ax+b\overline{x}=c, \qquad a,b,c\in E. $$ Under what condition does this equation have a solution in $E$, and when is it unique?

Answer 1

In a degree $2$ extension, everything looks like $u + vj$ for some chosen $j \in E$. So, write $a = a_1 + a_2j$, $x = x_1 + x_2j$, and so on. Rewrite $ax + b\bar{x} = c$ in terms of that, with $\bar{x} = x_1 - x_2j$. Expand all that stuff out and match coefficients. You end up with a matrix: $$ \begin{pmatrix} a_1 + b_1 & (a_2 - b_2)j^2 \\ a_2 + b_2 & a_1 - b_1 \end{pmatrix} $$ There is a unique solution when the determinant is nonzero. That's when $a\bar{a} \neq b\bar{b}$.