Does there exist a group homomorphism $(\mathbb{Q}_p, +) \to (\mathbb{R}, +)$?

Question

There exist homomorphisms from the $p$-adic numbers to the multiplicative group of real numbers (additive and multiplicative characters), but is there an additive homomorphism?

Answer 1

As mentioned in the comments, as $\mathbb Q$-vector spaces (and hence also as abelian groups), $\mathbb Q_p\cong\mathbb R$. Thus, there are many group homomorphisms between them.

This "morally" should not be the case, since $\mathbb Q_p$ and $\mathbb R$ seem to have such different structures. One way to reflect this is to consider toplogy: there are no nontrivial continuous group homomorphisms $\mathbb Q_p\to\mathbb R$.

Indeed, let $\varphi\colon\mathbb Q_p\to\mathbb R$ be a continuous group homomorphism, and pick any integer $n\ge0$. Then, look at the restriction $\varphi\colon p^{-n}\mathbb Z_p\to\mathbb R$. Since $p^{-n}\mathbb Z_p$ is compact, its image under $\varphi$ must also be compact. But $\mathbb R$ doesn't have any compact subgroups other than $\{0\}$, so $\varphi(p^{-n}\mathbb Z_p)=\{0\}$. Since this is true for all $n$, we conclude $\varphi=0$.

There are no continuous group homomorphisms $\mathbb R\to\mathbb Q_p$ either, which I will leave for you to prove.