Is the vector space $\mathbb R$ over $\mathbb Z/2\mathbb Z$ isomorphic to $\mathbb Z/2\mathbb Z$? How about $\mathbb R$ over $\mathbb Q$?

Question

From Hoffman & Kunze Linear Algebra:

Is the vector space $\mathbb R$ over $\mathbb Z/2\mathbb Z$ isomorphic to $\mathbb Z/2\mathbb Z$

Is the vector space $\mathbb R$ over $\mathbb Q$ isomorphic to $\mathbb Q$?

I am trying to think about this but it is a bit confusing. What is a basis for $\mathbb R$ over $\mathbb Z/2\mathbb Z$? Can it be $\{1\}$?

If I try to let $\{1\}$ be a basis then the only elements I can get are $0$ and $1$ since I must have $z\cdot 1$ for $z \in \mathbb Z/2\mathbb Z$.

Answer 1

1. A vector space structure would require a ring homomorphism $\mathbb Z/2\mathbb Z\to \operatorname{End}_\mathbb Z(\mathbb R)$, but the only homomorphism is zero, since the identity of the field of two elements has to map to something with finite additive order.

2. Is the vector space $\mathbb R$ over $\mathbb Q$ isomorphic to $\mathbb Q$? You are apparently asking "isomorphic as $\mathbb Q$ vector spaces." Of course not. One is $1$ dimensional, the other is infinite dimensional.