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The Local-Global Principle

cut-the-knot.org · 926 words · saved by 1 readers

The Local-Global Principle that was discovered in the 1920s by Helmut Hasse (so it is also known as the Hasse Principle) was the first major discovery that pointed to the utility of p-adic numbers. The set Q of the rational numbers is a (topological) field which is expanded to either R, the field of the reals, or to the various fields Q p, depending on the norm used. The expansion fields are quite different, but each expands Q algebraically, meaning that, for any pair of rational numbers r and s, the results of the operations r ± s, r×s, and r/s, being rational numbers, are preserved as such in all the expansions. So that, for example, r + s = t, if true in Q, is also true in R and all Q p. The terminology is this. Q is said to be a global field while all its expansions, R included, are said to be local. Thus any relation between a number of rational numbers which is true globally (i.e., in Q) is also true locally, i.e., in R and all the Q p's. The Global-Local Principle asserts a part

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