Fermat's Last Theorem and the Projective Curve $C_N$

Question

In "Silverman & Tate" on page 230 in the appendix on projective geometry, there is the remark:

The $N$th Fermat curve $C_N$ is the projective curve:

$$C_N: X^N + Y^N = Z^N$$

and Fermat's Last Theorem asserts that $C_N (\mathbb{Q})$ consists of only those points where one of $X$, $Y$, or $Z$ equal to zero.

I would appreciate help understanding how this equivalence comes about.

From what little I have read in the preceding pages, if $Z=0$ then there is a point at infinity, but even in that regard, I am uncertain of its implication for FLT.

Thanks.

Answer 1

Suppose a rational solution $(x,y,z)$. Let $d$ be a common denominator of $x,y,z$. Then $(dx)^n + (dy)^n= (dz)^n$ and we have an integral solution corresponding to it.