P-adic numbers complete/incomplete

Question

P-adic numbers are complete in one sense and incomplete in another sense. Is it so?

Firstly, does not complete mean connected? I read somewhere that there is not intermediate value theorem for p-adics because they are not connected. (if I am correct).

It seems I need elaboration of this "It can be shown that the rationals, together with the $p$-adic metric, do not form a complete metric space. The completion of this space can therefore be constructed, and the set of $p$-adic numbers $\mathbb{Q}_p$is defined to be this completed space."Math World.

How is the completion constructed with same metric and same numbers?

Answer 1

To see that $\mathbb Q$ is incomplete under the $p$-adic valuation, it suffices to find an element of $\mathbb Q_p$ not in the rationals. For $p=2$, $\sqrt{-7}$ will do, for $p=3$, $\sqrt7$ will do, and for $p>3$, the field $\mathbb Q_p$ contains all $p-1$ roots of unity of order $p-1$. The existence of these irrationalities in the $p$-adics drops out of Hensel’s Lemma, the basic fact of $p$-adic life.

Answer 2

After re-reading your question I think I understand what you want explained: take $\mathbb{Q}$, define the $p$-adic absolute value on it as $\left| x \right|_p = p^{-n}$, when $p$ is a factor in the unique factorization of $x$ ($ x = p^n a/b$, and $p$ does not divide $a$ or $b$), if $p$ is not part of the unique prime divisors as above, we say that the absolute value with respect to that is $1$.

Now we define a metric on the rationals as follows: $d(x,y) = \left| x - y \right|_p$. With respect to this metric the rationals are not complete (I can't come up with an example though) and I believe what I've read. We now take the completion of the rational with respect to this metric. The following is the general way to do it:

Let's take all Cauchy sequences with value in $X$ and let's call this set $M$. An element $x \in M$ is a sequence $x_n$; let us define $d(x,y) = \lim d(x_n.y_n)$ for all $x,y \in M$. This is not a metric, but having distance $0$ is an equivalence relation and if we take $M / \sim$, where $\sim$ is "having distance $0$", then we have a metric space with the metric defined above.

We now need to recover the points of $x$ as Cauchy sequences, in order to find $X \subseteq M/\sim$. The most logical way to do this is consider a point $x \in X$ as a sequence $x_n \in M$ converging to $x \in X$. Once we do this, it is clear that $X \subseteq M/\sim$.

If you do this for $\mathbb{Q}$ and the metric above defined, then you have $\mathbb{Q}_p$.

The question about completeness and connectedness is answered in my above comment. And yes, you're right: the intermediate value theorem relies on the connectedness of the space, thus the $p$-adic numbers do not have that property.