How do I Factor Odd and even Degree polynomial as a product of irreducible polynomial?

Question

I want to factor $x^n -1$ Into a product of irreducible polynomials over The Reals, when n is Odd and when n is even.

I know that The only irreducible polynomials over The Real are first Degree and second Degree polynomials.

But im stuck, so any hints would be great

Answer 1

The linear factors are easy because the real roots of $x^n -1$ can only be $\pm 1$.

Irreducible quadratic factors come from complex roots. The complex roots of $x^n -1$ are the $n$-th roots of unit: $\omega^k$, where $\omega=\exp(\frac{2\pi}{n} i)$. These roots come in conjugate pairs: $\omega^k, \bar\omega = \omega^{n-k}$. The quadratic polynomial having them as roots has real coefficients and is irreducible.

Full solution:

Linear factors:

When $n$ is odd, the only linear factor of $x^n -1$ is $x-1$.

and

When $n$ is even, the only linear factors of $x^n -1$ are $(x-1)(x+1)$.

Quadratic factors:

The quadratic factors of $x^n -1$ are $x^2-2Re(\omega^k)+1$ for $k=2,\dots,n-1$.