This is my first post here, so first of all, Congrats on this website, it seems very instructive and usefull!
I m currently studying p-adic number fields for my master degree and have a problem I can't seem to resolve.
Consider the p-adic local field F=Q_3 and its 2 extensions k1=Q_3(z_13) and k2=Q_3(z_3), when z_n denotes a primitive n-th root of unity.
Since (3, 13) = 1, k1 is unramified over F and k2 is obviously totally ramified over F.
Moreover, [k1:F] = 3 is the order of 3 in the multiplicative group Z_13* since 3^3=27=13*2+1.
on the other hand [k2:F] = 3-1 = 2.
Both are cyclic sub-extensions of k1k2 / F
Now, consider the extension k1k2 / k2. This is a cyclic extension of degree 3 and by ramification theory, its unramified.
Now by Kummer theory, since z_3 is in k2 its generated by some root a^(1/3) for some a in k2. We can assume a is integral.
But now, reducing mod P, when P is the maximal ideal of k2 over (3), the equation x^3 = a (mod P) has solutions in the residue field of k2 since the power-3 map is just the frobenius automorphism on the residue field of k2. Hence the residue field extension corresponding to k1k2 / k2 is trivial and then k1k2 is totally ramifie over k2! That's a contradiction...
So there is abviously a mistake somewhere in my "proof" but I can't find out where it is...
I could really use some help,
thanks a lot