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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 $K_x$:

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

Many thanks in advance

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