Does there exist a canonical projection from Z^p-1 to Z_p

[NoodleDurh]

Okay, I thought that there would exist a projection from $\mathbb{Z}^{p-1}$ to $\mathbb{Z}_{p}$ where $p$ is a prime. Like there exist a projection from $\mathbb{Z}$ to $\mathbb{Z}_p$

 

[jgens]

Assuming Z_p denotes the integers modulo p (and not the p-adic integers), then there is a "natural" projection Z^p-1→Z/p obtained simply by looking at the coproduct of the p-1 natural maps Z→Z/p.

 

[NoodleDurh]

okay, yeah you assumption is correct. Also, not to get to far off topic, but what does the p-adic integers look like.

 

[jgens]

Also, not to get to far off topic, but what does the p-adic integers look like.

Just consider the inverse limit of the sequence ...→Z/p³→Z/p²→Z/p and this produces the p-adic integers. Another way is completing Q with respect to the p-adic norm and then consider all element with p-adic norm less than or equal to one.