SCHWEITZER 1967 Problem 1

Let $$f(x)=a_0+a_1x+a_2x^2+a_{10}x^{10}+a_{11}x^{11}+a_{12}x^{12}+a_{13}x^{13} (a_{13} \not=0)$$ and $$g(x)=b_0+b_1x+b_2x^2+b_{3}x^{3}+b_{11}x^{11}+b_{12}x^{12}+b_{13}x^{13} (b_{3} \not=0)$$ be polynomials over the same field. Prove that the degree of their greatest common divisor is at least $6$ .

L. Redei