minimal polynomial of nth root of 2 over $\mathbb Q$

Question

The minimal polynomial of nth root of unity is the cyclotomic polynomial. What is the minimal polynomial of nth root of 2? Is it related to the cyclotomic polynomial?

Answer 1

By Eisenstein's Criterion applying to $x^n-2$, we find that all of the coefficients other than the leading coefficient are either $0$ or $-2$, meaning all of the coefficients other than the leading coefficient is divisible by the prime $2$. Also, the constant is not divisible by $2^2$. Therefore, Eisenstein's Criterion tells us that $x^n-2$ is irreducible, so the minimal polynomial for the $\text{n}^\text{th}$ root of $2$ must be $x^n-2$.