# Some number theory qual problems

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.

1. 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 $K(\zeta_\infty)$.

2. 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?)

3. 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)

4. 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 $K\otimes\mathbb{Q}_p\simeq \prod_{\mathfrak{p}\mid p}K_\mathfrak{p}$?)

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On item 3: Isn't it so that the Galois group of $\mathbf{Q}(\zeta_{7^n})$ has a unique subgroup of index 3? – Jyrki Lahtonen Jan 3 '12 at 9:23

(1) Taking into account Kronecker--Weber, this is the same as asking whether or not $G_K^{ab} \to G_{\mathbb Q}^{ab}$ is injective. If you are allowed to use class field theory, then look at the corresponding map on the idelic side. Is it injective?
(2) You could talk about how the Artin map depends on the splitting behaviour of this polynmoial mod $p$.
(3) $K$ is a degree $3$ extension of $\mathbb Q$. The Galois group of $\mathbb Q(\zeta_{7^n})$ over $\mathbb Q$ is $(\mathbb Z/7^n)^{\times}$. (For a more general discussion of how to make Kronecker--Weber effective, see e.g. this post.
(4) This cubic has disriminant equal to $-31$, and in fact generates the Hilbert class field of $\mathbb Q(\sqrt{-31})$. The splitting of prime ideals in the Hilbert class field is governed by a certain rule. Use this to describe $K\otimes \mathbb Q_p$.