What Does a = b (mod pZp) Mean in p-adic Numbers?

[pablis79]

Hi all,

If Zp is the ring of p-adic integers, what does the notation a = b (mod pZp) mean ? I understand congruence in Zp, i.e., a = b (mod p) implies a = b +zp, where z is in Zp (and a, b in Zp). However, I don't get what is meant by (mod pZp) ... does this mean a = b (mod p^k) for all k >= 1 ?

Thanks,
P

 

[micromass]

The p-adic numbers can be written in the form

$$\sum_{i=0}^{+\infty}{a_ip^i}$$

for $0\leq a_i\leq p-1$.

The ideal $p\mathbb{Z}_p$ is the ideal generated by p. It contains elements like

$$\sum_{i=1}^{+\infty}{a_ip^i}$$

Now, we say that $x=y~(mod~p\mathbb{Z}_p)$ if $x-y\in p\mathbb{Z}_p$.

 

[pablis79]

Ok, thanks micromass. So put another way it means that x and y are congruent modulo p in Zp.

Cheers!

 

[zetafunction]

have another question , if $$p \rightarrow infty$$ , how can you prove that the infinite prime $$p= \infty$$ is just the hole of the Real numbers ??

 

[morphism]

have another question , if $$p \rightarrow infty$$ , how can you prove that the infinite prime $$p= \infty$$ is just the hole of the Real numbers ??

As phrased, this question makes no sense.

I suspect what you're asking is, "Why do people say that $\mathbb Q_p = \mathbb R$ when $p=\infty$?" This is more a matter of convention (and convenience) than anything. There is no "let p -> infinity" going on. What is going on is that Q has several absolute values: up to equivalence, these are the p-adic absolute values (|.|_p) and the usual absolute value (|.|). One then completes Q at these absolute values to obtain the fields Q_p and R, respectively. One says "Q_p is the completion at of Q at p". Then there are good reasons to think of the usual absolute value as coming from an "infinite" prime, and to say that "R is the completion of Q at the infinite prime".