Exercise
Does the binomial theorem, i.e., the formula for the expansion of (a + b)m, also hold in residue class rings?
Yes, all the familiar
arithmetic rules also hold for residue class rings, and this is all that
is needed.
No, e.g., in R = Z/pZ you cannot divide by
k! if k is greater than p (since then k! = 0).
No, sometimes you get
unexpected cancellation as in (X2 + 1)2 =
X4 + 2X2 + 1 = 2X2 + 1
in the residue class ring Q[X]/(X3).
Good.
Wrong, just don't divide by k!.
No, terms may disappear, but this is no unexpected cancellation.