Find a real number t, and two polynomials F(X) and G(X) in Z[X] (integer polynomials ring) such that:
F(t)=2^(1/2)
G(t)=3^(1/2)
I have seen other problems of the same kind and I have not idea how to solve them.
With this I tried to find a polynomy of odd degree with only odd exponents and a t=a2^(1/2)+b3^(1/2) but I didn't find one who meets the conditions and I was randomly seeking for it.

Question

Find a real number t, and two polynomials F(X) and G(X) in Z[X] (integer polynomials ring) such that:
F(t)=2^(1/2)
G(t)=3^(1/2)
I have seen other problems of the same kind and I have not idea how to solve them.
With this I tried to find a polynomy of odd degree with only odd exponents and a t=a2^(1/2)+b3^(1/2) but I didn't find one who meets the conditions and I was randomly seeking for it.

anonymous · Answer

Other similar (to me) problem:
Find a polynomial:
P(X)/ P(X) is divisible for x^2+1 and P(X)+1 is divisible for x^3+x^2+1