Deduce the rational root theorem from Gauss’ Lemma

Use Gauss’ lemma to prove the Rational Root Theorem: If R is a Euclidean domain with fraction field Q and if p(x) = \sum_{i=0}^n c+i x^i is a polynomial in R[x], then any root of p(x) in Q has the form \frac{a}{b} where a|c_0 and b|c_n.


Suppose \frac{a}{b} is a root of p(x) in Q in lowest terms (i.e. a and b are relatively prime). Now 0 = \sum_{i=0}^n c_i \frac{a^i}{b^i}, so that (multiplying both sides by b^n) 0 = \sum_{i=0}^n c_i a^i b^{n-i}.

Now -c_0 b^n = \sum_{i=1}^n c_ia^ib^{n-i}. Since a divides each term on the right hand side, -c_0 b^n = at for some t. Since a and b are relatively prime, a and b^n are relatively prime. Thus a|c_0, by a previous exercise.

Similarly, -c_na^n = bs for some s, and we have b|c_n.

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