On the local Kronecker-Weber theorem (Q1813642)

Without using local class field theory, the author proves the local Kronecker-Weber theorem, that is, \(K\subset k_{ur}k_ \pi\) for any abelian extension \(K/k\) of a local field \(k\), where \(k_{ur}\) is the join of unramified extensions of \(k\), \(k_ \pi\) the field obtained by adjoining to \(k\) the roots of any composition \(f\circ f\circ\cdots\circ f\) of \(f=x^ q+\pi x\), \(\pi\) any prime element of \(k\), \(q=\#k\pmod \pi\). Properties of Galois cohomology are used.