Definition 2

Let $\mathscr{S}$ be the set of maximal ideals of L. We define

Br(L$/K$) $=\bigcap_{\mathfrak{m}\in\mathscr{S}}$ Ker($r_{(L/\mathfrak{m})/K}$), where $r_{(L/\mathfrak{m})/K}:$ Br(K) $\rightarrow$ Br(L$/\mathfrak{m}$) is the extension of scalars homomorphism (VIII, p. 281).