TITLE: How to solve polynomial equations in a field and/or in a ring?
QUESTION [18 upvotes]: I'm studying for my exam, and I stuck on solving polynomials in a field and/or in a ring. Let me give you some examples:
(1) Solve equation $x^2+4x+3=0$ in field $\mathbb{Z}_5$, $\mathbb{Z}_8$ and in ring $\mathbb{Z}_{12}$ and $\mathbb{Z}_5 \times \mathbb{Z}_{13}$
(2) Solve equation $x^2+2x=1$ in field $\mathbb{Z}_7$, $\mathbb{Z}_{11}$ and in ring $\mathbb{Z}_{12}$
(3) Find ring $\mathbb{Z}_n$, in which equation $x^2+2x=3$ has at least 3 solutions.
I'm not asking for solutions for these particular problems. I want to understand this topic, so I would appreciate example solutions of similar problems. Also, you're aware of some study materials - please list tutorials, books etc., about solving polynomial equations in different rings and fields (and similar topics).
Thanks for help.

REPLY [5 votes]: HINT $\rm(3)\:\ $ diagonalize employing Chinese remainder: $\rm\:(m,n)=1\ \Rightarrow\ \mathbb Z/mn \cong \mathbb Z/m\times \mathbb Z/n\:,\:$ hence $\rm\:(x-1)\:(x+3)\:$ has root $\rm\:(1,-3) \ne (1,1),(-3,-3)\in \mathbb Z/m \times \mathbb Z/n\ $ for $\rm\ m,n \ne\cdots$
NOTE $\ $ The ring $\rm\:D\:$ is a domain iff every polynomial $\rm\ f(x)\in D[x]\ $ has at most $\rm\ deg\ f\ $ roots in $\rm\:D\:.\: $  For a simple proof see my post here, where I illustrate it constructively in $\rm\ \mathbb Z/m\ $ by showing that, given any $\rm\:f(x)\:$ with more roots than its degree, we can quickly compute a nontrivial factor of $\rm\:m\:$ via a $\rm\:gcd\:$. The quadratic case of this result is at the heart of many integer factorization algorithms, which attempt to factor $\rm\:m\:$ by searching for a nontrivial square root in $\rm\: \mathbb Z/m\:,\:$ e.g. a square root of $1$ that is not $\:\pm 1$.