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](https://en.wikipedia.org/wiki/Lagrange_inversion_theorem) would allow me to examine series solutions with a special case being the Bring ultraradicals.



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


