Embedding into $p$-adic complex numbers

Question

As I'm reading notes about the Leopoldt conjecture, the following question came to my mind:

Let $\mathbb{C}_p$ be the $p$-adic complex numbers, i.e. the completion of the algebraic closure of the field of $p$-adic numbers $\mathbb{Q}_p$. For a number field $K$ there are embeddings $\sigma: K \to \mathbb{C}$ and Dirichlet's Unit Theorem determines the structure of the units of this number field $\mathcal{O}_K^\times$ via a logarithm map. Now, to construct the same for the $p$-adic complex numbers, one takes the logarithm map $\log_p: k^\times \to k$ which is defined for a local field $k$ via $\log_p(p) = 0$ and the convergent series $\log_p(1+x) = x - x^2/2 + ...$ for every principal unit $1+x \in U^1_k$.

Now there is a map

$$K^\times \to \prod_{\sigma \in Hom(K,\mathbb{C}_p)} \mathbb{C}_p^\times$$

that I do not understand. Where do the embeddings into the $p$-adic complex numbers come from? I know that the $p$-adic complex numbers and the complex numbers itself are isomorphic as abstract field, however, not as topological fields.

Can somebody help me with this question? Thank you!

Best, Tom

Answer 1

I don't know anything about the Leopoldt conjecture, but regarding embeddings into $\mathbb{C}_p$.... Note that $K$ is a finite extension of $\mathbb{Q}$, and $\mathbb{C}_p$ is an algebraically closed field containing $\mathbb{Q}$, so there have to be embeddings of $K$ into $\mathbb{C}_p$. Abstractly this is the same way you get embeddings of $K$ into $\mathbb{C}$. To be more explicit, you can use the primitive element theorem to pick a primitive element $\theta$, so that $K = \mathbb{Q}(\theta)$, and then the minimal polynomial $p(x)$ of $\theta$ over $\mathbb{Q}$ must split over $\mathbb{C}_p$ since $\mathbb{C}_p$ is algebraically closed. If $\theta_1, \dotsc, \theta_n$ are the roots of $p(x)$ in $\mathbb{C}_p$, then $\theta \mapsto \theta_i$ define the $n$ possible embeddings of $K$ into $\mathbb{C}_p$.

Answer 2

Fix an embedding of $C_p$ into $C$. Then any embedding of K into $C_p$ becomes an embedding of K into $C$ , and that's all, no topology is involved. Regarding Leopoldt's conjecture at p, the approach via p-adic log in order to construct the p-adic regulator works only for totally real fields. For general number fields, there is a more convenient conceptual approach. Let $E_1$ denote the group of units of K which are congruent to 1 mod every prime $v$ of K above p. Let $U_1,v$ be the local units congruent to 1 mod $v$ . By the diagonal embedding of $E_1$ into the direct sum of the $U_1,v$ 's, the closure of $E_1$ becomes a $Z_p$-module, and Leopoldt's conjecture states that the $Z_p$-rank of this closure is equal to the $Z$-rank of $E_1$. The best reference for all this, in my opinion, is Washington's book "Introduction to cyclotomic fields", §§5.5, 13.1 and 13.5.