UTM Ideals Varieties and Algorithm - Chapter 1 Section 2 Exercise 11

Problem:

Solution:

The two trivial solutions for both odd and even powers are

$x = 0$, $y = 1$
$x = 1$, $y = 0$

The two more trivial solutions for only even powers are

$x = 0$, $y = -1$
$x = -1$, $y = 0$

Suppose we have a non-trivial solution in $\mathbf{Q}$, suppose the answer is $x = \frac{p}{q}$, $y = \frac{r}{s}$, then we have

$(\frac{p}{q})^n + (\frac{r}{s})^n = 1$

 Multiply both side by $q^ns^n$, we get a non-trivial solution for Fermat's last theorem.

$(ps)^n + (rq)^n = (qs)^n$.