A field extension of degree 2 is a Normal Extension.

Question

Let $L$ be a field and $K$ be an extension of $L$ such that $[K:L]=2$. Prove that $K$ is a normal extension.

What I have tried : Let $ f(x)$ be any irreducible polynomial in $L[x] $ having a root $\alpha$ in $K$ and let $\beta$ be another root. Then I have to show $\beta \in K$. I already have $L(\alpha)=K$, but now how to show $\beta\in K??$ Any hints?

Answer 1

Since $\alpha \in K$ is a root of the irreducible polynomial $f \in L[X]$, then $f$ is the minimal polynomial of $\alpha$ over $L$. The degree $d$ of $\alpha$ over $L$ is $≤2$, because $[K : L]=2$.

1. If $d=1$, what can you conclude?

2. If $d=2$, write $f(X)=X^2+aX+b=(X-\alpha)(X-\beta)$. What are the relations between $\alpha$ and $\beta$?

If $d=1$, then $f(X)=X-\alpha$ so that $\beta=\alpha \in K$. If $d=2$, then $X^2+aX+b=X^2-(\alpha+\beta)X+\alpha\beta$ so that $-a=\alpha+\beta$, or $\beta = -a-\alpha \in K$, since $a \in L \subset K$.