I've been going through the following list of problems:
http://www.ocf.berkeley.edu/~mgsa/wiki/Number_Theory
Most of them are routine and some are more or less algebraic geometry (I haven't looked at those), but there are a few that I don't know how to do. I've solved most of them except for the following. I'm hoping someone could point me in the right direction or even show how they're done. I've written in parenthesis some comments on what I've done on each problem.
Given a number field $K$ which is not $\mathbb{Q}$. Prove there exists an abelian extension $L/K$ such that $L$ is not contained in $K(\zeta_\infty)$.
Let $f=X^3-X^2-2X+1$, $\alpha$ a root of $f$ and $K=\mathbb{Q}(\alpha)$. If a prime $p$ is unramified in $K$, what does this mean about the Artin map corresponding to $K$? (Could this mean anything in addition to the fact that it's defined and that the Frobenius map is independent of the prime above $p$ since the extension is Abelian? Is there something else that's important?)
Let $f=X^3-X^2-2X+1$, $\alpha$ a root of $f$ and $K=\mathbb{Q}(\alpha)$. Find a cyclotomic extension of $\mathbb{Q}$ containing $K$. (The discriminant is 49, so the extension is abelian, so some $\mathbb{Q}(\zeta_{7^n})$ will do, since the conductor is of the form $7^n$, but I don't know how to bound $n$ i.e. how to compute the conductor)
Suppose that $-31$ is not a square modulo $p$, where $p$ is prime. What can you say about $K\otimes \mathbb{Q}_p$, where $K=\mathbb{Q}(\alpha)$ and $\alpha^3+\alpha+1=0$? (Is there anything more except that $K\otimes\mathbb{Q}_p\simeq \prod_{\mathfrak{p}\mid p}K_\mathfrak{p}$?)