Quadratic extensions with elementary abelian \(K_ 2(O)\) (Q1178925)

Suppose that \(F\) is a totally real algebraic number field with exactly one dyadic prime, and let \(S\) be the set of infinite primes of \(F\) together with the dyadic prime. Suppose also that \(F\) has odd \(S\)-class number and that \(F\) contains \(S\)-units with independent signs. Under these hypotheses, the author gives necessary and sufficient conditions for the 2-primary part of the Milnor group \(K_ 2(O_ E)\) to be elementary abelian, where \(E\) is a quadratic extension of \(F\). Some explicit computations are given.