Question

I know less than you know in the topic since you are a teacher now.
However, I want to mention two sources you can find the proof which use little prerequisites of algebra.
First of all, the classic Basic Nuber Theory by Andre Weil contains a proof in the first section of the first chapter which uses a great method.
As for the second, the Chinese mathematician Hua, Lo-keng ( in Chinese 華羅庚 ) has published a book entitled Introduction to number theory, which has a proof that uses only elementary terminologies, and I hope it is exactly what you need.
By the way, the first approach is the same as that mentioned by @QiaoChu Yuan in some sense, and the second is mostly elementary.
P.S. I hope it is acceptable here using Chinese, thanks.