Quick help finishing off a proof: extension of p-adic fields

[simba31415]

Homework Statement

Let $L_1/K,\,L_2/K$ be extensions of p-adic fields, at least one of which is Galois, with ramification indices $e_1,\,e_2$. Suppose that $(e_1,\,e_2) = 1$. Show that $L_1 L_2/K$ has ramification index $e_1 e_2$.

Homework Equations

I have most of the proof done: I'm trying to show $e_1 e_2 | e$ where $e$ is the ramification index of $L_1 L_2/K$, and then show $e | e_1 e_2$. It's easy enough to show that $e_1 e_2 | e$: if we define $e_1 '$ to be the ramification index between $L_2, \, L_1 L_2$ and likewise $e_2 '$ between $L_1, \, L_1 L_2$, then $e = e_1 e_2' = e_2 e_1'$ by multiplicative property of ramification indices. So both $e_1,\,e_2$ divide $e$ and are coprime so $e_1 e_2 | e$.

All I need now is to show that $e_1' | e_1$, then $e_1' e_2 = e | e_1 e_2$ (or alternatively to show $e_2' | e_2$). However, I can't seem to manage this: indeed quite the opposite, I keep deducing $e_1 | e_1'$ and $e_2 | e_2'$ by using the 2 forms of $e$ and coprimality. I also, as far as I know, haven't used the fact one of the extensions is Galois (or that they are of p-adic fields) yet, unless I have forgotten a precondition for one of the results I've used. Could anyone help complete my proof? Thanks! -S

 

[morphism]

I would try to show that $e\leq e_1e_2$. If L_1/K is Galois, what kind of inequality does that impose on e_1?

 

[simba31415]

I would try to show that $e\leq e_1e_2$. If L_1/K is Galois, what kind of inequality does that impose on e_1?

Hi morphism, sorry it took me so long to reply - I didn't see anyone had responded to me. I've been having a think, but I can't think of any obvious inequalities which we can deduce for e_1: presumably we want $e_1 \geq$ something for this. I know all the ramification indices for any prime splitting downstairs must be the same in this case, but I don't think that helps us here.

So it would suffice to show that $e_1 \geq e_1'$, but while I could believe it I don't know if I could prove it. Do you think you could give me another hint perhaps? Sorry, it's probably something very obvious I'm overlooking!