Exploring Embedding $\epsilon_p: \mathbb{Z} \to \mathbb{Z}_p

[evinda]

Hello! (Smile)

We have the canonical function $\epsilon_p: \mathbb{Z} \to \mathbb{Z}_p, x \mapsto (\overline{x})_{k \in \mathbb{N}_0}=(\overline{x}, \overline{x}, \overline{x}, \dots )$.
The function $\epsilon_p: \mathbb{Z} \to \mathbb{Z}_p$ is an embedding. We interpret with it $\mathbb{Z}$ as a subset of $\mathbb{Z}_p$.

Proof:
Let $x \in \mathbb{Z}$ with $\epsilon_p(x)=(\overline{x})_k=0$, i.e. $x \equiv 0 \mod{p^{k+1}}$ for all $k \in \mathbb{N}_0$. The only integer number that is divisible by any high $p$-power is zero, so it holds $x=0$. Thus, $\epsilon_p$ is an embedding.

Could you explain me the above proof? (Thinking)

 

[jvicens]

Hello! Sure, I'd be happy to explain the proof for you.

First, let's define what an embedding is. An embedding is a function that maps one mathematical structure (in this case, the integers) to another structure (in this case, the integers modulo p) in such a way that preserves the structure and properties of the original structure.

Now, let's break down the proof step by step.

1. The function $\epsilon_p: \mathbb{Z} \to \mathbb{Z}_p$ is defined as $\epsilon_p(x)=(\overline{x})_k=(\overline{x}, \overline{x}, \overline{x}, \dots )$. This means that for any integer x, $\epsilon_p$ maps it to the sequence of its residue classes modulo p (i.e. the remainder when divided by p).

2. To prove that $\epsilon_p$ is an embedding, we need to show that it preserves the structure and properties of the integers. In other words, if we apply $\epsilon_p$ to an integer, the resulting sequence of residue classes should still behave like an integer in terms of divisibility and other properties.

3. Now, let's consider an integer x that maps to the sequence $(\overline{x})_k=0$. This means that for all values of k, $x \equiv 0 \mod{p^{k+1}}$, which essentially means that x is divisible by all higher powers of p.

4. However, the only integer that is divisible by all higher powers of p is 0. This can be easily seen by considering the prime factorization of any integer. Therefore, if $x \equiv 0 \mod{p^{k+1}}$ for all k, it must be that x=0.

5. This proves that $\epsilon_p$ maps the integer x to 0 if and only if x=0. This is exactly what we need to show in order for $\epsilon_p$ to be an embedding - it preserves the property of being zero.

I hope this helps clarify the proof for you! Let me know if you have any other questions.