In this post, I’d like to explain a proof of the Law of Quadratic Reciprocity based on properties of Lucas polynomials. (There are over 300 known proofs of the Law of Quadratic Reciprocity in the literature, but to the best of my knowledge this one is new!)
In order to keep this post as self-contained as possible, at the end I will provide proofs of the two main results (Euler’s criterion and the Fundamental Theorem of Symmetric Polynomials) which are used without proof in the body of the post.
The sequence of Lucas polynomials is defined by , , and for
The next few terms in the sequence are , and .
By induction, the degree of is equal to . The integers are the Lucas numbers, which are natural “companions” to the Fibonacci numbers (see, e.g., this blog post).
It’s easy to see that for odd, is divisible by and has only even-power terms. Thus for some monic integer polynomial of degree . We will be particularly interested in the polynomials for prime.
If is even (resp. odd), a simple induction shows that the constant term (resp. the coefficient of ) in is equal to . In particular, for odd we have .Continue reading