- #1

NoodleDurh

- 23

- 0

## Homework Statement

Show that if ##|\mathbb{Z}_p/(f)| = p^{deg(f(x))}## where the ##deg(f(x)) = n## and ##f(x) \in \mathbb{Z}_{p}## is irreducible over ##\mathbb{Z}_{p}[x]##, then ##\mathbb{Z}_{p}[x]/(f) \rightarrow_{\phi}^{\cong} \bigoplus_{i \in I} \mathbb{Z}_{p_{i}}## where ##|I| = n##

## Homework Equations

## The Attempt at a Solution

(An attempt at rigor)

Surjection is trival. Obviously, the size of ##\bigoplus_{i \in I} \mathbb{Z}_{p_{i}}## and ##\mathbb{Z}_{p}[x]/(f)## are the same, so ##\bigoplus_{i \in I} \mathbb{Z}_{p_{i}}## is covered by ##\mathbb{Z}_{p}[x]/(f)## completely, and thus ##im( \phi ) = \bigoplus_{i \in I} \mathbb{Z}_{p_{i}}##. Injectivity is less so. Since ##\phi## is a homomorphism we can see that ##\phi (a) = \phi (b)## iff ## \phi (a) - \phi (b) = 0## when ##\phi (a-b) = 0##, precisely when ##a-b \in ker (\phi ) ## hence, ##a = b##, So now we must check the order of the elements. (this is where I am stuk)

Last edited: