Suppose that $F$ is a field. Show that there exists a $F$-division algebra $D$ with two elements $a\neq b\in D$ such that $a^2-2ab+b^2=0$.
In the field extensions we know that $a^2-2ab+b^2=0$ if and only if $a=b$, because of $a^2-2ab+b^2=(a-b)^2$. But I know this is not true if the extension is division ring, but I can't prove it.
In a part of my research I want two elements like that in order to extend $F$ to $F(a)$ such that there exists $b\notin F(a)$ such that we know its minimal polynomial $p(x)=x^2-2ax+a^2$.
It is a little part of my research and I don't have any idea to prove or simplify this question. About 3 years ago one of my professors told me this is true but now, he isn't in touch.