On p-adic L-functions and cyclotomic fields

Let p be a prime. If one adjoins to Q all pn-th roots of unity for n = 1, 2, 3, …, then the resulting field will contain a unique subfield Q∞ such that Q∞ is a Galois extension of Q with Gal the additive group of p-adic integers. We will denote Gal(Q∞/Q) by Γ.