Greatest common divisor of $X^n-1$ and $X^m-1$

Question

Let $f=X^n-1$ and $g=X^m-1$ be two polynomials. Show that: $$\left(f,g\right)=X^{\left(n,m\right)}-1,$$ where $\left(a,b\right)=$ greatest common divisor of $a$ and $b$.

Answer 1

Suppose m and n have a common divisor like k such that $m=m_1k$ and $n=n_1k$ then we can write:

$x^n-1=x^{n_1k}-1=(x^k-1)(x^{n_1k-k}+x^{{n_1k-2k}}+ \ldots+ {x^{n_1k-(n_1-1)k}}+1)$

$x^m-1=x^{m_1k}-1=(x^k-1)(x^{m_1k-k}+x^{{m_1k-2k}}+ \ldots +{x^{m_1k-(m_1-1)k}}+1)$

Which their common divisor is..$(f,g)=(x^k-1)$ or $(f,g)=x^{gcd(m,n)}-1$