# Relation between tame symbol and residue on a curve

For an discrete valuation field $$K$$ we can define the tame symbol:

$$(\,,\,)_K:K^\times\times K^\times\to \overline K^\times$$ $$(a,b)\mapsto(-1)^{v(a)v(b)}\overline{a^{v(b)}b^{-v(a)}}$$

Consider now a smooth projective curve $$X$$ over a finite field $$k$$ and let $$K=K(X)$$. We denote with $$(\,,\,)_x$$ the tame symbol for the complete discrete valuation field $$K_x$$.

For any element in the space of differential forms $$K_xdt$$ we have also the notion of residue. Suppose that

$$\omega:=dt\sum a_it_i\in k(x)((t))\,.$$ then we define $$\operatorname{res}_x(\omega):=a_{-1}\in k(x)$$.

What is the relation between $$(\,,\,)_x$$ and $$\operatorname{res}_x$$?

The both satify reciprocity laws: Weyl reciprocity law, and Tate reciprocity law. So I guess that we can express:

$$(a,b)_x=\operatorname{res}_x(f(a,b))\,,$$ where $$f:K_x^\times\times K_x^\times\to K_x^\times dt$$ is "a function". Is this true, can we calculate explicitly the tame symbol by using the residue? For example we know that for $$a$$ in the invertible part of the local ring of $$K_x$$:

$$(a,t)_x=\operatorname{res}_x\left(\frac{a}{t}dt\right)$$