Okay, I think I finally understand the problem.  For some reason I thought we were taking the product over all roots also.  Let's try one more time:

The field $\mathbb{Q}(\sqrt{-7})$ has class number $1$.  The prime $2$ splits into the distinct primes $(1/2)(-1\pm \sqrt{-7})$.

Let $f(x)=2x^{2}+x+1$.  The roots of this polynomial are $(1/4)(-1\pm \sqrt{-7})$, which both have no places $\nu$ where the valuation is positive.  (Scratch that.  This polynomial is a unit.)===== I'm going to go and think harder about this problem.