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The tame symbol serves many purposes in mathematics, and is of particular value when we use it to evaluate curves over certain number _elds. A well-known example is that of the Hilbert symbol, which gives us insight into the existence of a rational solution to a conic of the form ax2 + by2 = c for a; b; c 2 Q_. In order to properly examine this symbol, we must gain a solid understanding into the p-adic completion of the rationals, Qp. We will explore the algebraic construction of the subring of p-adic integers, Zp, whose _eld of fractions is Qp itself. In general, we may look at a type of tame symbol, which we call a local symbol, that we take over an algebraic curve defined over a field into some abelian group G. The properties of these local symbols correspond directly to those of the Hilbert symbol. We then examine if it is possible to de_ne a type of local symbol over a degree 2 extension of Z, the Gaussian Integers Z[i]. The construction of this symbol is analogous to one for a degree 2 extension of Z which is a Euclidean domain. Central extensions of groups are explored to give a foundation for viewing the tame symbol in terms of determinates as viewed by Parshin.
