A remark on the positivity of \(K_{2}\).
Let \(F\) be a number field. The Hilbert symbols at the real infinite primes of \(F\) induce a split surjective homomorphism \[ K_2(F) \rightarrow \bigoplus_{v \,\text{real}}\{\pm1\}, \] whose kernel is the positive \(K_2\)-group \(K_2^+(F)\), introduced and first studied by \textit{G. Gras} [J. Number Theory 23, 322--335 (1986; Zbl 0589.12010)]. \(K_2^+(F)\) is generated by all symbols \(\{u,v\}\) with \(u,v\) totally positive. The authors generalize this result as follows: Given \(m\) distinct real embeddings \(\mu_1,\dots,\mu_m\) of \(F\) the kernel of the homomorphism \[ K_2(F) \rightarrow \bigoplus_{i=1}^m \{\pm 1\} \] induced by the Hilbert symbols at the embeddings \(\mu_i\) is generated by all symbols \(\{u,v\}\) with \(u,v\) positive at the given embeddings \(\mu_1,\dots,\mu_m\). The result extends previous work of \textit{R. V. Moody} and \textit{J. Morita} [J. Algebra 229, 1--24 (2000; Zbl 0956.19001)] on some relative \(K_2\)-groups associated with a positive cone.
scientific article; zbMATH DE number 2037071