- #1
MostlyHarmless
- 345
- 15
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.
...Define ##v_p: ℤ-0\rightarrowℝ## then for each integer ##n≠0## let ##v_p(n)## be the unique positive integer satisfying ##n=p^{v_p(n)}n'## with p not dividing n'...
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.