The real numbers are a field extension of the rationals? In preparing for an upcoming course in field theory I am reading a Wikipedia article on field extensions. It states that the complex numbers are a field extension of the reals. I understand this since $\mathbb R(i) = \\{ a + bi : a,b \in \mathbb R\\}$. Then the article states that the reals are a field extension of the rationals. I do not understand how this could be. What would you adjoin to $\mathbb Q$ to get all the reals? The article doesn't seem to say anything more about this. Is there a way to explain this to someone who has yet to take a course in field theory?

Saying "the reals are an extension of the rationals" just means that the reals form a field, which contains the rationals as a subfield. This does _not_ mean that the reals have the form $\mathbb{Q}(\alpha)$ for some $\alpha$; indeed, they do not. You have to adjoin _uncountably many_ elements to the rationals to get the reals.