The discussion revolves around proving that the polynomial equation x^(6) - x^(2) + 2 = 0 has no constructible roots. Participants explore the nature of the roots and the implications of constructibility in the context of algebraic equations.
The discussion is ongoing, with participants offering different perspectives on how to approach the problem. Some suggest using properties of constructible numbers and algebraic orders, while others express uncertainty about the next steps in their reasoning.
There is a mention of the need to show that the polynomial has no rational roots, and participants are considering the implications of irrational roots in relation to constructibility. The definitions and assumptions regarding constructible numbers are also being questioned.
saadsarfraz said:prove that x^(6) - x^(2) +2 =0 has no constructible roots
saadsarfraz said:ok i made the substitution x=y+1 and now I have this polynomial y^6 + 6y^5 + 15y^4 + 20y^3 + 14y^2 + 4y + 2 = 0, all i have to do is show that this has irrational roots but i just don't know how.