TITLE: Let $K$ be a subfield of $\mathbb{C}$ not contained in $\mathbb{R}$. Show that $K$ is dense in $\mathbb{C}$.
QUESTION [9 upvotes]: Let $K$ be a subfield of $\mathbb{C}$ not contained in $\mathbb{R}$. Show that $K$ is dense in $\mathbb{C}$.


completely stuck on it. can I get some help please.

REPLY [7 votes]: Hint: Show that the (topological!) closure of $K$ is a subfield $L$ of $\mathbb{C}$.  Since $K$ contains $\mathbb{Q}$, $L$ contains $\mathbb{R}$, the topological closure of $\mathbb{Q}$.  Since $[\mathbb{C}:\mathbb{R}] = 2$ and $L$ is assumed to contain an element of $\mathbb{C} \setminus \mathbb{R}$, we must have $L = \mathbb{C}$.