1
$\begingroup$

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?

1 Answers 1

4

Hint $\: $ Mimic the proof of the Rational Root Test, which works over any domain where gcds exist, i.e $\rm\:(a,b) = 1,\ 0 = b^n\:\! f(a/b) =\: a^n + b\:(\cdots)\ \Rightarrow\ b\:|\:a^n\ \Rightarrow\ b\:|\:1\:$ by Euclid's Lemma.

  • 0
    Too tru$e$, my memory was faulty. The order of proofs is e$x$actly the other way round (and it seems that Gauss' Lemma is around 45 years older).2012-04-10