so there's the following fact: For all finite fields $K$, it holds that $$\prod\limits_{k \in K^*} k = -1.$$ I know how to prove it, but I'd really like to just include a reference. However, for the life of me I somehow cannot find this fact as a theorem or similar in any of my Algebra books. Can anybody tell me whether this fact has a name, or give me a reference to some book that proves it?
Thanks!
The product of an invertible element with its inverse is $1$. If we just break this product up into pairs of elements $x, x^{-1}$, then the product of each pair will be $1$, so the whole product should be $1$, right?
This argument fails precisely when $x = x^{-1}$. When is that? $x^2 - 1$ is a degree $2$ polynomial, so it has at most two roots. Indeed, they are precisely $+1$ and $-1$. So the product breaks up into one $1$, one $-1$, and a bunch of pairs that multiply to $1$.