For each n \ge 1 define \sM_n = \text{maximal abelian } p\text{-extension of } F_n \text{ unramified outside } \mathfrak{p}_n, \sL_n = \text{maximal unramified abelian } p\text{-extension of } F_n, and analogously \sM_n^+, \sL_n^+ over F_n^+ (unramified outside \mathfrak{p}_n^+). Passing to the limit, \sM_\infty = \bigcup_n \sM_n, \quad \sL_\infty = \bigcup_n \sL_n, \quad \sM_\infty^+ = \bigcup_n \sM_n^+, \quad \sL_\infty^+ = \bigcup_n \sL_n^+, so that \sM_\infty is the maximal abelian pro-p-extension of F_\infty unramified outside \mathfrak{p}, and \sL_\infty is the maximal unramified abelian pro-p-extension of F_\infty.

Define the Galois groups \sX_\infty = \Gal(\sM_\infty/F_\infty), \qquad \sX_\infty^+ = \Gal(\sM_\infty^+/F_\infty^+), \sY_\infty = \Gal(\sL_\infty/F_\infty), \qquad \sY_\infty^+ = \Gal(\sL_\infty^+/F_\infty^+), using the extensions of Definition 12.2.1. By construction the fields sit in towers F_n \subseteq \sL_n \subseteq \sM_n and F_\infty \subseteq \sL_\infty \subseteq \sM_\infty (and likewise with the {}^+ superscripts), so \sY_\infty = \Gal(\sL_\infty/F_\infty) is a quotient of \sX_\infty = \Gal(\sM_\infty/F_\infty). The modules \sX_\infty^+, \sY_\infty^+ are the analogous Galois groups over the totally real tower F_\infty^+.

The modules \sX_\infty, \sY_\infty are \Lam(\GG)-modules, and \sX_\infty^+, \sY_\infty^+ are \Lam(\GG^+)-modules. For x \in \sX_\infty and \sigma \in \GG, lift \sigma to any \tilde\sigma \in \Gal(\sM_\infty/\Q) and set \sigma \cdot x := \tilde\sigma\, x\, \tilde\sigma^{-1}. This is well defined because \sX_\infty is abelian, so the conjugation is independent of the lift. Since \cO_L[\GG] is dense in the Hausdorff ring \Lam(\GG), the action extends by \cO_L-linearity and continuity to all of \Lam(\GG). The actions on \sY_\infty, \sX_\infty^+, \sY_\infty^+ are defined identically. This refers to Definition 12.2.2 and the Iwasawa algebra Definition 3.3.3.