Clear[PrimitiverTeil]

PrimitiverTeil[a_, x_]:=Module[
{liste, nenner, lcm, gcd, k},
liste = CoefficientList[a, x];
nenner = Map[Denominator, liste];
lcm = LCM[Apply[Sequence, nenner]];
liste = liste*lcm;
gcd = GCD[Apply[Sequence, liste]];
liste = liste/gcd;
Sum[
              liste[[k]] x^(k - 1), {k, 1, Length[liste]}]
]

a = (∑_(k = 0)^10) Random[Integer, {0, 9}]/Random[
                Integer, {1, 9}] (x^k)

b = (∑_(k = 0)^10) Random[Integer, {0, 9}]/Random[
                Integer, {1, 9}] (x^k)

c = PrimitiverTeil[a, x] PrimitiverTeil[b, x] // Expand

GCD[Apply[Sequence, CoefficientList[c, x]]]