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

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?

• According to WolframAlpha, the minimal polynomial seems to be just $x^n-2$. Mar 27 '16 at 13:27

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 $x^n-2$.