# Let $L:K$ be a Galois extention, show that $L:M$ is a normal.

Assume the field extension $L:K$ is Galois (i.e. finite, normal and separable), with $M$ an intermediate field.

• Show that $L:M$ and $M:K$ are finite separable field extensions.

Attempt: Both $M:K$ and $L:M$ are algebraic ($M:K$ since $M \subset L$, so every $\alpha \in M$ is algebraic over $K$ and $L:M$ since any $\alpha \in L$ is algebraic over $K$, so also over $M$).

Then $M:K$ is separable again since $M \subset L$ (every element of $L$ so in particular every element of $M$ is separable over $K$).

For $L:M$, take any $\alpha \in L$, then min$_M(\alpha)$ divides min$_K(\alpha)$ in $M[X]$. Since min$_K(\alpha)$ has no multiple zeros in a splitting field, neither does min$_M(\alpha)$, i.e. $\alpha$ is separable over $M$.

I'm not quite about the proof for finite, would it involve the Tower Law?

• Show that $L:M$ is a normal extension.

I'm a bit stuck here, any help for this one?

• Your proof will depend on the definition of normality that you’re using. What’s yours? – Lubin Jun 2 '18 at 19:37
• @Lubin "A field extension $L:K$ is normal if every irreducible polynomial over $K$ which has at least one zero in $L$ splits in $L$." – user12002 Jun 2 '18 at 21:28
• Take an $M$-irreducible polynomial $f$ with a root $\alpha$ in $L$. Look at the minimal (irreducible) $K$-polynomial for $\alpha$, call it $g$. What do you know about $f$ versus $g$? (You might look at some examples.) – Lubin Jun 3 '18 at 1:56

Yes, to show that $$L/M$$ and $$M/K$$ are finite extensions use the Tower Law, $$[L:K] = [L:M][M:K].$$
To show that $$L/M$$ is a normal extension, let $$f \in M[x]$$ be an irreducible polynomial which has a root $$\alpha$$ in $$L$$. We want to show that $$f$$ splits over $$L$$. Let $$g \in K[x]$$ be the minimal polynomial of $$\alpha$$ over $$K$$. Then, $$f$$ divides $$g$$. Since $$L/K$$ is normal, $$g$$ splits over $$L$$ and therefore so does $$f$$.