Field inclusion

Question

In my book about $p$-adic numbers, inclusion $\mathbb{Q} \hookrightarrow \mathbb{Q_{p}}$ is mentioned quite a lot of times, but I have no idea what this actually means. So, what exactly does a field inclusion mean? And specifically, what does the mentioned inclusion represent?

Answer 1

There is no need for anyone to go read this just to give you essentially the same answer that Jyrki already has.

The book calls this an inclusion because it has constructed $\Bbb Q_p$ in such a way that $\Bbb Q$ is not an actual subset of it. But there is a subfield of $\Bbb Q_p$ which is isomorphic to $\Bbb Q$ (as is true of any field of characteristic 0). That isomorphism from $\Bbb Q$ to the subfield of $\Bbb Q_p$ is the inclusion being referred to.

The subfield in $\Bbb Q_p$ is identical with $\Bbb Q$ in every sense except that the actual elements in it are not the same set-theoretic objects as the elements in $\Bbb Q$. But it still has a $0$ and a $1$, and from these, you can use induction to show that it has an equivalent of the natural numbers. Negation and division then extend this to all rational numbers.

There is no reason to make a big deal about this subfield being distinct from $\Bbb Q$. In fact, it is quite possible to use a different construction of $\Bbb Q_p$ so that $\Bbb Q$ actually is the subfield, not just isomorphic to it. Most authors therefore take no notice of the distiction, and simply call the subfield $\Bbb Q$, no matter how they have constructed $\Bbb Q_p$.