Prove certain properties of the function $d_p(a,b)=p^{-v_p(a-b)}$

Question

I was ill and didn't manage to attend a few classes on number theory. Now, I'm struggling to prove a theorem that was presented during the lectures I didn't addend and intentionally ,,left as an exercise'' by the lecturer:

define function $d_p:\mathbb{Z}\times\mathbb{Z}\rightarrow \mathbb{R}$ such that $d_p(a,b)=p^{-v_p(a-b)}$, where $v_p$ is the $p$-adic exponent of $a - b$. Prove that the following properties are true:

• $d_p(a,b)=0\Leftrightarrow a-b=0$
• $d_p(a,b)\ge 0$
• $d_p(a,c)\le d_p(a,b)+d_p(b,c)$
• $d_p(a,b)=d_p(b,a)$

I'm sorry to say the only thing I thought of was writing numbers $a$ and $b$ in the following form: $a=p^{\alpha}\frac{m}{n}$ and $b=p^{\beta}\frac{z}{x}$. Then, I guess, we have:

$$ a-b=p^{\alpha}\left(\frac{m}{n}-p^{\beta-\alpha}\frac{z}{x}\right)$$

but I don't know if it's of any use in proving any of the four properties.

I'm not looking for a complete solution - I know you guys don't want to do all the work for me, but could somebody drop me at least some hint?

Answer 1

Let me address the properties in order. Note that you must also note that you're taking the convention that $p^{-\infty} = 0$, since $v_p(0) = \infty$ (if you have left $v_p(0)$ undefined, you'll need to separately define $d(a,a) = 0$ for any $a$). Here are some leading questions/suggestions that should help you out. Below, I'll say "$p$ divides $n$ $m$ times" to mean that $n = p^m n'$, where $(n',p) = 1$; in other words, I'm saying that $v_p(n) = m$.

1. You essentially want to show that $v_p(x) = \infty$ if and only if $x = 0$. If $n$ is a nonzero number, can $p$ divide $n$ infinitely many times?
2. Can $p^x$ be negative for any real value of $x$? What if "$x = -\infty$"?
3. I would recommend proving something stronger: that $d_p(a,c)\leq \max(d_p(a,b),d_p(b,c))$. Translate this into a statement about the $v_p$'s: you should find that you want to show $v_p(a-c)\geq\min(v_p(a-b),v_p(b-c))$. Now, set $a - b = x$, $b - c = y$. Then this statement is equivalent to $$v_p(x + y)\geq\min(v_p(x), v_p(y)).$$ Since $x$ and $y$ are integers, you can write $x = p^{v_p(x)} x'$, $y = p^{v_p(y)} y'$ for some $x',y'\in\Bbb Z$ relatively prime to $p$. Now, consider how many times $p$ can divide their sum. First suppose that $v_p(x)\geq v_p(y)$ without loss of generality, and then use your thought above.
4. You know that $v_p(n)$ is the number of times $p$ divides $n$. How many times does $p$ divide $-n$?