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Let $L/K/\mathbb Q$ be a tower of number fields. The result I want to show is that the discriminant $\Delta_K$ divides the discriminant $\Delta_L$.

I was wondering if there was a "direct" proof of this just by using the trace representation (or anything else). For example, following the procedure of this question gives $$ [L : K]^2 \Delta_K = c^2 \Delta_L $$ for some integer $c$, which doesn't quite give what I want, but I'm curious whether with some modifications this could be made to work.

Alternatively, by noting that any rational prime which ramifies in $K$ also ramifies in $L$, we get that any prime dividing $\Delta_K$ also divides $\Delta_L$, which is again a "near miss". If this line of thought could be made to work, I'd love to see it, too.

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Evan Chen
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    Have you looked at the proof in Neukirch's book? He shows that the discriminant is obtained by taking the norm of the different. Then since the different and norm are multiplicative in towers of extensions, the result follows. But I'm not sure that you would call this direct. – Stiofán Fordham Apr 28 '15 at 15:18

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