jmc

Question

Find a nonzero monic polynomial P(x) with integer coefficients and minimal degree such that P(1-[3]2+[3]4)=0. (A polynomial is called monic if its leading coefficient is 1.)

Accepted Answer

Let x = 1 - [3]2 + [3]4. Note that (1 - [3]2 + [3]4)(1 + [3]2) = 3, so x = 31 + [3]2.Then 3x = 1 + [3]2,so 3x - 1 = 3 - xx = [3]2.Cubing both sides, we get -x^3 + 9x^2 - 27x + 27x^3 = 2,so -x^3 + 9x^2 - 27x + 27 = 2x^3. This simplifies to 3x^3 - 9x^2 + 27x - 27 = 3(x^3 - 3x^2 + 9x - 9) = 0, so we can take f(x) = x^3 - 3x^2 + 9x - 9.