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?

For $\mathbb{R}$ to be field extension of $\mathbb{Q}$, all we need is that $\mathbb{R}$ is a field containing $\mathbb{Q}$ as a subfield. That's definitely true.
The construction is a bit more delicate and analytic in nature: $\mathbb{R}$ is the completion of $\mathbb{Q}$ and is substantially larger. Since $\mathbb{R}$ is uncountable, it's not an extension of finite degree, meaning that you cannot write $\mathbb{R} = \mathbb{Q}(a_1, a_2, ..., a_n)$ for some finite sequence of symbols $a_i$ (nor even a countable sequence). You have to adjoin uncountably many symbols.

If you're interested in a way to construct reals from rationals, take a look at Dedekind cuts. 
