My professor mentioned during lecture that the roots of $x^3-2=0$ over $\mathbb{Q}$ are $\sqrt[3]{2}, j\sqrt[3]{2}, j^2\sqrt[3]{2}$, where $j$ is a root of $x^2+x+1$.
I would like some clarification on the connection between the roots of $x^3-2=0$ and the roots of $x^2+x+1$. What is the significance of $x^2+x+1$ and why do two of the roots of the irreducible polynomial $x^3-2=0$ over $\mathbb{Q}$ depend on it?
How does the polynomial $x^2+x+1$ change when we consider roots of irreducible polynomials of higher degree? (i.e. $x^n-k=0$ for $n,k \in \mathbb{Z}$)
Thank you
For a square free integer $a$ the solutions to $X^p-a$ are $b\zeta^k$ for $0 \leq k \leq p-1$ where $\zeta$ is a primitive p-th root of unity and $b$ is the p-th root of $a$. So in this case $b=\sqrt[3]{2}$ and $\zeta$ is a primitive 3rd root of unity.