## Every subfield of the real numbers must contain the rational numbers

Prove that any subfield of $\mathbb{R}$ must contain $\mathbb{Q}$.

By the previous exercise, $\mathbb{R}$ contains a unique inclusion-smallest subfield which is isomorphic either to $\mathbb{Z}/(p)$ for a prime $p$ or to $\mathbb{Q}$.

Suppose the unique smallest subfield of $\mathbb{R}$ is isomorphic to $\mathbb{Z}/(p)$, and let $a$ in this subfield be nonzero. Then $pa = 0$ in $\mathbb{R}$, and since $p \in \mathbb{R}$ is a unit, $a = 0$, a contradiction.

Thus the unique smallest subfield of $\mathbb{R}$ is isomorphic to $\mathbb{Q}$. In particular, any subfield of $\mathbb{R}$ contains a subfield isomorphic to $\mathbb{Q}$.