By T.S. Bhanumurthy
Compatible for college students and academics of arithmetic, this ebook bargains with the historic continuity of Indian arithmetic, ranging from the Sulba Sutras of the Vedas as much as the seventeenth century.
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Extra info for A Modern Introduction to Ancient Indian Mathematics
An interesting case is the following. Let G be a group and (Hn) a decreasing sequence of normal subgroups of G. We can then take Gn = G/Hn and (since 1. p-adic Numbers 32 cp : G/H the canonical projection homomorphism. The C projective limit of this sequence is a subgroup of the product G= lim G/H C O G/H E-together with the restrictions of projections ifr : G --* G/H,,. Since the system of quotient maps f, : G -p- G/H is always a compatible system, we get a factorizaG such that f = lJr o f. It is easy to determine the kernel of this tion f : G factorization f : f,,=nH,,.
5. Additive Structure of Qp and Zp Let us start with the sum formula Qp = Zp + Z[1/p] proved in the last section. Observe that this sum is not direct, since Zp n Z[1/p] = Z. fro oar The various embeddings that we have obtained are gathered in the following commutative diagrams giving the additive (resp. multiplicative) structure of Qp (resp. Qp ). Q Qp Z[1/p] Z[l/p] Zp QX QX P Zx (p) P z Z xP T N z P T N (1) (1) If we embed Z in the direct sum Zp ® Z[1/p] by means of m F-;, (m, -m) and call r the image, then the addition homomorphisms Z(p) ® Z[l/P] Z(p) + Z[l/P] = Q, Zp ® Z[l/p] - Zp + Z[l/p] = Qp have kernel IF and furnish isomorphisms (Z(P) (D Z[1/P]) / r = Q, (ZP ® Z[l/P]) / IF = Q.
Hence s = 1h, - hl E H and,-, -* 0. Since H is an additive subgroup, we must also have Z - 8 C H (for all n > 0), and the subgroup H is dense in R. 3. Topological Algebra 23 Corollary. (a) The only proper closed subgroups of R are the discrete subgroups aZ (a E R). C1+ (b) The only compact subgroup of R is the trivial subgroup (0}. (f' Using an isomorphism (of topological groups) between the additive real line and the positive multiplicative line, for example an exponential in base p R -+ R>0 t i-+ p`, (the inverse isomorphism is the logarithm to the base p) we deduce parallel results for the closed (resp.