# Math Help - Fields

1. ## Fields

let a= cuberoot(2+sqrt(6)), what does the field Q(a) look like?
If B=a^2 +a +2, then express B^-1 in the form alpha_5(a^5)+ alpha_4(a^4)+...+alpha_0 where the alpha_i's exist in Q.

2. Originally Posted by Coda202
let a= cuberoot(2+sqrt(6)), what does the field Q(a) look like?
If B=a^2 +a +2, then express B^-1 in the form alpha_5(a^5)+ alpha_4(a^4)+...+alpha_0 where the alpha_i's exist in Q.
If $a=\sqrt[3]{2+\sqrt{6}}$ then $a^3 - 2 = \sqrt{6}$ so $a^6 - 2a^3 - 2 = 0$. Thus, $a$ is root of polynomial $x^6 - 2x^3 - 2$. This polynomial is irreducible so $\mathbb{Q}(a)$ are all elements of form $a_0 + a_1 a+ ... + a_5a^5$ for $a_i \in \mathbb{Q}$.