# Number Theory: Ramification

I am currently trying to figure out the following question regarding ramification.

Let K = $\mathbb{Q}(\sqrt{5})$, L = $\mathbb{Q}(\sqrt{7})$, M = $\mathbb{Q}(\sqrt{35})$, and KL = $\mathbb{Q}(\sqrt{5},\sqrt{7})$. Show that KL/M is unramifed.

From what I understand, the example able would be a counter-example of Dedekind's Discriminant Theorem if M is not $\mathbb{Q}$. The professor hinted that transitivity needed to be applied to show that KL/M is unramifed. What I have attempted to do is find the ramification indices of prime numbers in the extensions L$/\mathbb{Q}$, K$/\mathbb{Q}$, M$/\mathbb{Q}$. Doing this, how would I eventually apply Transitivity?

• Did you try computing the discriminant of $KL$ over $\mathbf{Q}$? – David Loeffler Oct 28 '14 at 17:18