Proof involving field extensions

[daon]

I have to prove that all quadratic extensions are normal. I think my problem here is the wording.

In my mind this is what it is asking: If f(x) is an irreducible quadratic polynomial over a field F, and f(x) splits in E, then the extension E/F is normal.

Would that be right?

So for example $\displaystyle f(x)=x^2+1$ is irreducible over $\displaystyle \mathbb{R}$, but since f(x) splits in $\displaystyle \mathbb{C}$, $\displaystyle \mathbb{C}$/$\displaystyle \mathbb{R}$ is normal.

 

[daon]

Okay, I am given that TFAE, which makes any extension E/F normal:

1) E is a splitting field of some polynomial f(x) in F[x]
2) For every algebraic extension K/E and ever F-homomorphism $\displaystyle \phi:E \rightarrow K, \,\, \phi(E)=E$
3) For every algebraic extension K/E and ever F-homomorphism $\displaystyle \phi:E \rightarrow K, \,\, \phi(E) \le E$ ($\displaystyle \le$ meaning subfield)
4) Every irreducible polynomial p(x) in F[x] which has a root in E, splits in E.

(2 and 3 seem redundant to me, but they're there.)

So I was thinking about letting $\displaystyle p(x)=ax^2+bx+c \in F[x]$ be irreducible and showing #4... that if $\displaystyle p(x)$ has a root in E, then both roots of $\displaystyle p(x)$ must be in E.

If $\displaystyle p(x)$ has a multiple root, then its trivial, so assuming p(x) separable (having no multiple roots). So if $\displaystyle E/F$ is an extension with $\displaystyle \alpha \in E$ a root of p(x) then $\displaystyle p(x)=(x-\alpha)(\beta x + \gamma) \in E[x]$.

Then I must have that $\displaystyle \beta$ and $\displaystyle \gamma$ both be in E and therefore the second root is $\displaystyle \frac{-\gamma}{\beta} \in E$. That seems too easy and I think I'm missing something.