Algebraic analysis

No. $C_\mathbb{Q}/ Z_\mathbb{Q} \cong \mathbb{R}$ and $\{ \{q\}_{n>0} + Z_\mathbb{Q} | q \in \mathbb{Q} \} \cong \mathbb{Q}$ . But $\mathbb{R}$ is not an algebraic extension of $\mathbb{Q}$ .

Adding to Jonathans solution, $\mathbb R$ is isomorphic to ${C_\mathbb Q}/{\mathbb Z_\mathbb Q}$ because there is a surjective homomorphism $f$ from $C_\mathbb Q$ to $\mathbb R$ whose kernel is $\mathbb Z_\mathbb Q$ . $f$ exists because the completion of $\mathbb Q$ is $\mathbb R$ and thus by the first isomorphism theorem there is an isomorphism from ${C_\mathbb Q}/{\mathbb Z_\mathbb Q}$ to $\mathbb R$