I have these two questions about the polynomial $f(X)=X^n-1 \in \mathbb{R}[X]$.
- Assume $n$ is even. Show that $f(X)$ has two real roots. Factorise $f(X)$ into a product of irreducible polynomials in $\mathbb{R}[X]$.
It is pretty easy to see that since $n$ is even $1,-1$ are the roots. My problem is with the factorization. I know that to factorize we have to have $f(X)=g(X)h(X)$, where $g(X),h(X)$ needs to be constants or units. I also know that $(X - a)(X - a^*) = X^2 - 2aX + a^2 + b^2$, where $a^*$ is the complex conjugate of $a$, but I'm unsure how to proceed.
- Assume $n$ is odd. Show that $f(X)$ has $1$ real root. Factorise $f(X)$ into a product of irreducible polynomials of $\mathbb{R}[X]$.
Again the same problem. The root is $1$, but don't know how to go about with the second part.