Steinberg symbol

In mathematics a Steinberg symbol is a pairing function which generalises the Hilbert symbol and plays a role in the algebraic K-theory of fields. It is named after mathematician Robert Steinberg.

For a field F we define a Steinberg symbol (or simply a symbol) to be a function ${\displaystyle (\cdot ,\cdot ):F^{*}\times F^{*}\rightarrow G}$, where G is an abelian group, written multiplicatively, such that

The symbols on F derive from a "universal" symbol, which may be regarded as taking values in ${\displaystyle F^{*}\otimes F^{*}/\langle a\otimes 1-a\rangle }$. By a theorem of Matsumoto, this group is ${\displaystyle K_{2}F}$ and is part of the Milnor K-theory for a field.

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