Let D be a UFD with quotient field F. If f (x) $\in$ D[x] is monic and b$\in$ F such that f (b) = 0 , then show that b$\in$ D.
All I know is
F is a field and f(b) = 0 therefore $f(x) = (x-b)q(x)$ and also f(x) $\in$ D[x] so f(x) can be written as the product of some irreducible elements of D[x].
What should I do to show that b$\in$ D?