proof of Pythagorean triples
Suppose that where . (here is the norm), so if and only if . is cyclic over with Galois group isomorphic to , so by Hilbert’s Theorem 90, there is some element such that
so that
Now, given any integer right triangle with , we have
where , so for some ,
Clearing fractions on the right hand side of these equations by multiplying numerator and denominator by the square of the least common multiple of the denominators of , we get
for . Thus for some ,