By Richard Courant, Charles de Prima, John R. Knudsen

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**Example text**

1). With respect to multiplication, all subgroups 1 + p"Zp (n > 1) are topological groups. (3) The real line R is an additive topological group. If a topological group has one compact neighborhood of one point, then it is a locally compact space. If a topological group is metrizable, then it is a Hausdorff space and has a countable fundamental system of neighborhoods of the neutral element. ' Let G be a metnzable topological group. Then there exists a metric d on G that defines the topology of G and is invariant under left translations: d(gx, gy) = d(x, y).

It defines a canonical isomorphism Z[[X ]]/(X - p) ZP, where (X - p) denotes the principal ideal generated by the polynomial X - p in the formal power series ring. PROOF Let us consider the sequence of homomorphisms fn : Z[[X]] -* Z/pnZ, >2aiX` - >aip` mod p". * limZ/p"Z = ZP compatible with the fn. If x = Y_ ai pi is any p-adic integer, then x = f (Y- ai X'), and this shows that f is surjective. We have to show that the kernel of f is the principal ideal generated by the polynomial X - p. In other words, we have to show that if the formal power series Y_ ai X' is such that Fi

Indeed, any rational number has the form x = p° (v E Z, a and b prime to p). When v = -m < 0, namely when x V Ztpt, we can use the Bezout theorem to express the fact that pm and b are relatively prime, (pm, b) = 1 = apm + Pb; 5. The Field Qp of p-adic Numbers 43 hence multiplying by x yields a as Pa x = p'"b = b + Pm E Z(p) + Z[1/P]. This gives an elementary description of the decomposition Q = Z(P) + Z[l/P] induced by the decomposition Qp = Zp + Z[1/p]. 5. Additive Structure of Qp and Zp Let us start with the sum formula Qp = Zp + Z[1/p] proved in the last section.