Cyclotomic polynomials without using the zeros of $Y^n-1$

This note aims to construct an ``intrinsic’’ splitting field for the polynomial $Y^n-1$ over the rational field $\mathbf{Q}$, in a way that Gauss, Kummer, Kronecker and Bishop would have liked. Contrary to the usual presentations, our construction does not use any splitting field of $Y^n-1$ which would be given before demonstrating the irreducibility of the cyclotomic polynomial.