Good afternoon!
I am stuck thinking about something in ring theory and I would like to know if anyone could give me a nudge the right way.
I have been working on the polynomial ring $\mathbb{Z}_p[X]$, where $p$ is a prime number and $\mathbb{Z}_p = \mathbb{Z}/p\mathbb{Z}$. Now, say you are trying to quotient that ring by principal ideals generated by polynomials of a certain degree $k$.
I'm pretty sure $\mathbb{Z_p}[X]/(x^k) \simeq \mathbb{Z_p}[X]/((x+\alpha)^k)$ for any $\alpha \in \mathbb{Z}_p$. Because if you denote respectively: $I = (x^k)$, $J = ((x+1)^k)$, then every element of $R/I$ can be written $I + p(x)$, with $p(x) \in \mathbb{Z_p}[X]$, and similarly for $R/J$.
Then, it's easy to show that the homomorphism $\phi : R/I \to R/J : I + p(x) \mapsto J + p(x + \alpha)$ has $\{0\}$ kernel, which implies bijectivity (because $R/I$ and $R/J$ are equipotent).
Now... with a friend, we conjectured that maybe $R/K \simeq R/I$ when $K$ is generated by a polynomial with at least one repeated root (or a certain number, we don't really know). But we do not really know if that's true, and where we can proceed from there.
If anyone happens to have an idea, or know of a theorem that could help us, I'd be grateful!
Thanks :D
Well, if $f(x) = f_1^{e_1}(x)\cdots f_m^{e^m}(x)$ is a product of distinct prime powers, there is a natural ring isomorphism between ${\Bbb Z}_p[x]/\langle f(x)\rangle$ and $${\Bbb Z}_p[x]/\langle f_1^{e_1}(x)\rangle \times \ldots \times {\Bbb Z}_p[x]/\langle f_m^{e_m}(x)\rangle.$$ As a note consider the notion of Galois ring.