Construction of valuations from \(K\)-theory
This expository paper gives a unified presentation of some explicit constructions of certain valuations on a field. In particular, given a prime number \(p\), a field \(K\) of characteristic \(\neq p\), and a subgroup \(T\) of \(K^\times\) containing \((K^\times)^p\) and \(-1\), suppose \(T\) satisfies the two conditions (i) if \(x \in K^\times \backslash T\) and \(y \in T\backslash K^p\), then \(\{x,y\} \neq 0\) and (ii) if the cosets of \(x,y \in K^\times\) in \(K^\times /T\) are \( F_p\)-linearly independent then \(\{x,y\} \neq 0\), where \(\{x,y\}\) denotes an element in the second Milnor \(K\)-group \(K_2^M(K)\) of \(K\). Then a valuation ring \(O\) on \(K\) is constructed having non-\(p\)-divisible value group, residue characteristic \(\neq p\), and for which the principal units are \(p\)th powers. The approach is elementary in that it uses neither cohomology nor noncommutative division rings, nor is the existence of primitive \(p\)th roots of unity in the field \(K\) assumed. The cohomological structures used in many previous proofs of the existence of such a valuation are replaced by the second Milnor \(K\)-group.
scientific article; zbMATH DE number 1489500