# $p$-adic integers and valuation

Let $$p$$ be a prime, $$x \in \mathbb{Z}_p$$. I say that if $$x \in \mathbb{Z}_p$$ then $$\text{val}_p(x) \ge 0$$ as follows:

Let $$x = a_0 + a_1 p + a_2 p^2 + \dots$$. Then, $$\text{val}_p(x) \ge \text{min}(\text{val}_p(a_0), \text{val}_p(a_1), \dots ) \ge 0$$.

Now, I think that $$\text{val}_p(x) \ge 0 \implies x \in \mathbb{Z}_p$$ but could not prove it. I start by assuming $$\text{val}_p(x) \ge 0$$ and $$x\notin \mathbb{Z}_p$$. Since $$x\in \mathbb{Q}_p$$, say $$x= \dots + a_{-k} \cfrac{1}{p^k} + \dots$$ for some $$a_{-k} \in \{1,\dots,p-1\}$$. Then, we can write

$$x = \cfrac{1}{p^k}(\dots + a_0 + \dots)$$ and we have $$\text{val}_p(x) = \text{val}_p(\cfrac{1}{p^k}) \text{val}_p(\dots + a_0 + \dots) = -k + \text{val}_p(\dots + a_0 + \dots) \ge 0$$.

However, I stuck here. So, can we say

$$\text{val}_p(x) \ge 0 \implies x \in \mathbb{Z}_p$$

or not?

• What’s your definition of $\Bbb Z_p$ ? – Lubin Jun 29 at 23:52
• Series of the form $\displaystyle \sum_{k=0}^{\infty} a_k p^k$ where $0 \le a_k < p$. – Ninja Jun 30 at 0:05
• What is your definition of the valuation then? Because the way I define it you do not have to prove anything. – ThorWittich Jun 30 at 0:08
• Given $x \in \mathbb{Q}$, one can write $x = p^k \frac{a}{b}$ where $k$ is an integer, $p,a,b$ are relatively prime integers. Then, $val_p(x) = k$. – Ninja Jun 30 at 0:11
• Don’t try for a proof by contradiction. If $v_p(z)\ge0$, is not $z$ of form $\sum_0^\infty a_kp^p$? – Lubin Jun 30 at 0:11

Let $$x \in \mathbb{Q}_p$$. That means we can write $$x = \sum_{k \geq -m} a_kp^k,$$ where $$\text{val}_p(x) = -m$$. Now assume that $$\text{val}_p(x) \geq 0$$. That means $$x$$ has to be of the form $$x = \sum_{k \geq 0} a_kp^k,$$ which just means it is an element of $$\mathbb{Z}_p$$.
• The valuation of $x$ - the valuation of the sum - is greater than or equal to $0$. So it must be greater than something negative, then how can we say your argument then? – Ninja Jun 30 at 1:28
• I am not sure if understand your comment. If the valuation is bigger or equal to zero we know that the smallest index having s non-zero coefficient is $0$. Therefore we can start the series there and these elements are exactly the elements of the $p$-adic integers. – ThorWittich Jun 30 at 13:09