Let $F$ be a field and $n , m \in \mathbb{N}$. Then an element $\varepsilon \in \overline{F}$, being $\overline{F}$ an algebraic closure of $F$, is called a $n$-th root of unity if it is a root of the polynomial $X^n – 1 \in F[X]$. Now suppose that $n$ and $m$ are coprime numbers ($\gcd(n , m) = 1$) and $\varepsilon$ is $n$-th and $m$-th root of the unity. Can I state that $\varepsilon = 1$? If $F = \mathbb{C}$, then it is trivial because we have the formula
$$
X^n – 1 = \prod_{d\big|_n} {\Phi}_d(X)\mbox{,}
$$
being ${\Phi}_d$ the $d$-th cyclotomic polynomial in $\mathbb{C}[X]$. But this formula fails in an abstract field $F$, so is it false in general?
There are not $n$-th and $m$-th primitive roots of the unity, where $n$ and $m$ are coprime numbers.
field-theory
Best Answer
To prove that statement in general, here is the problem that you need to solve (by induction on $\max (n,m)$):
$$\gcd(x^n-1, x^m-1)= x^{\gcd(n,m)}-1$$
This holds over every field! Then your question follows immediately.