I need help in this proof.
The question is: Let be F a field. Suppose that $car(F) \not=2$, which $car(F)$ means the characteric and let be $K/F$ an extension with $[K/F]=2$. Prove that $K/F$ is a Galois extension.
I have no idea how to proceed. Any suggestions?
Normal:
Let $a$ be an element of $K$ not in $F$, $1,a$ is a $F$-basis of $K$, you have $a^2=ua+v$, it implies that $a$ is a root of $P(X)=X^2-uX-v$ we deduce that $K$ is the splitting field of $P$. Divide $P$ by $X-a$, you have $P(X)=(X-a)(X-b)+c$, since $P(a)=0$, we deduce that $P(X)=(X-a)(X-b)$.
Separable:
if $a$ in $K$ not in $F$ the minimal polynomial $P=(X-a)(X-b)$ of $a$ has degree $2$, $P'=2X-a-b$ $P'(a)=0$ implies that $a-b=0$, we deduce that $a=b$, $P=(X-a)^2=X^2-2a+a^2$ since the characteristic is not $2$, $2a\neq 0$ and is an element of $F$, we deduce that $a\in F$ contradiction, so $P'(a)\neq 0$.