Factor $\sum_{n=1}^{N} x^n$ [closed]

Question

I am attempting to factor an $N^{\text{th}}$ degree polynomial with coefficients strictly equal to $1$ given by the equation

$$\sum_{n=1}^{N} x^n$$

Although the Galois group for anything beyond a quartic is not generally soluble, I had hoped that an existing result had been established for this particular case. If not, I was curious if generalizing Tchirhausen transforms to the $N^{\text{th}}$ order and employing the Lagrange inversion theorem would allow me to examine series solutions with a special case being the Bring ultraradicals.

Answer 1

This is not a research level question, nevertheless I feel like answering it. If you divide the polynomial by $x$, and multiply by $x-1$, you get $x^N-1$. The latter factors over $\mathbb{Q}$ as the product of the cyclotomic polynomials $\Phi_d(x)$ with $d$ dividing $N$ (the factor corresponding to $d=1$ is $x-1$ that we added artificially). These factors are irreducible over $\mathbb{Q}$. Factoring over $\mathbb{R}$ is even easier, you just need to group the nonreal roots of $x^N-1$ (i.e. the $N$-th roots of unity except $\pm 1$) into complex conjugate pairs. Hope this helps!