# P-adic norm, valuation, and expansion

1. Feb 25, 2015

### MostlyHarmless

In this book I'm working from (P-adic numbers: An introduction by Fernando Gouvea), he gives an explanation of the p-adic expansion of a rational number which I'm pretty sure I understand this, but a bit later he talks about the p-adic absolute value, which from what I understand is the same as the norm. He defines the valuation like this:
Which I can understand that, my interpretation is that this is basically another way of decomposing an integer that emphasizes the prime factor that we care about. So $v_p(n)$ is just the exponent on p in the the prime factorization of n.

Then he discusses the p-adic absolute value, and in short he defines it as
$$|x|_p=\left\{ \begin{array}{l} |x|_p=p^{-\alpha}\text{ if } p^{\alpha}|x\\ |x|_p=1 \text{ if }p\not|\text{ } x \end{array} \right.$$

I'm a little foggy as to how these 3 things are related. I'm pretty sure he defines the valuation the way he does for the proof that $|x|_p$ is a metric. But I don't see how the expansions are related.

2. Mar 2, 2015