well, clearly, every multiple of p(t) is congruent to the 0-polynomial. and since we can write, for ANY polynomial f(t) in F[t]:
f(t) = p(t)q(t) + r(t), where the degree of r is less than the degree of p, it's clear that every element of F[t]/β can be written:
r(t) + B, where B is the equivalence class of 0 (and thus p(t)) under β. and since r has degree n-1 or less:
is a spanning set for F[t]/β. so dim(F[t]/β) ≤ n.
this implies p(t) divides ,
and since the latter polynomial has degree < degree p(t), it must be the 0-polynomial. that is:
, so dim(F[t]/β) ≥ n. that leaves just one possibility....