The aim of this work is to provide a new approach for constructing $n$-dimensional Steinberg symbols on discrete valuation fields from $(n+1)$-cocycles and to study reciprocity laws on curves related to these symbols.

Keywords:

Steinberg symbols, reciprocity laws, discrete valuation field, algebraic curves Steinberg symbols, reciprocity laws, discrete valuation field, algebraic curves

MSC Classifications:

19F15, 19D45, 19C09 show english descriptions Symbols and arithmetic [See also 11R37]
Higher symbols, Milnor $K$-theory
Central extensions and Schur multipliers 19F15 - Symbols and arithmetic [See also 11R37]
19D45 - Higher symbols, Milnor $K$-theory
19C09 - Central extensions and Schur multipliers

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