Formula for Hilbert symbol for primes lying above 2

Question

Let $K$ be any number field, and let $\mathfrak{p}$ be a prime lying over 2. Is there a formula for computing the Hilbert symbol $(a,b)_\mathfrak{p}$? I know the formula when $\mathfrak{p}$ lies above an odd prime, but I can't find anything in the case for the prime 2. Just to be clear, I'm interested in the general number field case (I know the formulas for $K = \mathbb{Q}$).

Answer 1

There is this very nice paper by John Voight:

http://www.math.dartmouth.edu/~jvoight/articles/quatalgs-060513.pdf

Chapter 6 has formulae for the dyadic Hilbert symbol.