I am trying to understand this proof That if L is splitting field of some polynomial then it's normal extension .
I got the part till we get that $L(\alpha)/K(\alpha)$ and $L(\beta )/K(\beta)$ are isomorphic as they are splitting field of same irreducible polynomial over two isomorphic fields ( right ? ) , on second page .
But after it how are we writing
$$[L(\alpha):K]=[L(\beta ):K]$$ ?
Why didn't we directly wrote $[L(\alpha):L]=[L(\beta ):L]$ ,as both $L(\alpha)$ and$L(\beta)$ are isomorphic and extension of L ?
Best Answer
We don't know a priori that $L(\alpha)$ and $L(\beta)$ are isomorphic extensions of $L$. This is exactly what we are trying to prove: we have that $L(\alpha) = L$, because we assume that $\alpha$ belongs to $L$, and we are trying to prove that any other root $\beta$ of the minimal polynomial $m \in K[x]$ of $\alpha$ over $K$ also belongs to $L$, so that $m \in K[x]$ must in fact split into linear factors over $L$.
Consider, for example, $K = \mathbb{Q}$, and $L = K(\sqrt[3]{2})$, and let $\alpha = \sqrt[3]{2}$. Then, minimal polynomial of $\alpha$ over $K = \mathbb{Q}$ is $m(x) = x^3 - 2$. The splitting field $M$ of $m$ is not $L$. Let $\beta$ be one of the complex roots of $x^3 - 2$. Then, $L(\alpha) = L$, but $L(\beta) \ne L$.
What might have confused you, though, is that we do easily know that $K(\alpha)$ and $K(\beta)$ are isomorphic extensions of $K$, because they are both isomorphic to the quotient ring $K[x]/(m)$ via maps $x \mapsto \alpha$ and $x \mapsto \beta$.