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

anonymous · Answer

For example I tried:
(2^(1/2)+3^(1/2))^(3)=
11*2^(1/2)+9*3^(1/2)
You also have:
(2^(1/2)+3^(1/2))^(1)=
1*2^(1/2)+1*3^(1/2)
(2^(1/2)+3^(1/2))^(5)=
109*2^(1/2)+89*3^(1/2)
And tried to write (1,0) or (0,1) as a combination of:
a(1,1)+b(11,9)+c(109,89)
But the difference between the first number and the second is always even, so the MCD is 2 and I need it to be 1 because I can't divide by 2 because Z[X] only has integer coefficients.
I tried a few ways of multipling 2 or 3 per something but the MCD was never 1 and I don't have idea if it even exist a couple of numbers a,b which:
(a2^(1/2)+b3^(1/2))^(2k+1), k>1 and a2^(1/2)+b3^(1/2) can be combined to write (1,0) using integer coefficients.
Sorry for my bad English but as no body answered I tried to make sure people know I tried to solve this before posting the question.

anonymous · Answer

Solved... I only had to divide my t for 2 and solve the problem of different dividend was solved multiplyng the coefficint of x^n for 1 Here we go: http://www.wolframalpha.com/input/?i=4t^3-9t%2Ct%3D%282^%281%2F2%29%2B3^%281%2F2%29%29%2F2 http://www.wolframalpha.com/input/?i=11t-4t^3%2Ct%3D%282^%281%2F2%29%2B3^%281%2F2%29%29%2F2