Number Theory
In number theory one may study a Diophantine equation, for example, modulo p for all primes p, looking for constraints on solutions. The next step is to look modulo prime powers, and then for solutions in the p-adic field. This kind of local analysis provides conditions for solution that are necessary. In cases where local analysis (plus the condition that there are real solutions) provides also sufficient conditions, one says that the Hasse principle holds: this is the best possible situation. It does for quadratic forms, but certainly not in general (for example for elliptic curves). The point of view that one would like to understand what extra conditions are needed has been very influential, for example for cubic forms.
Some form of local analysis underlies both the standard applications of the Hardy-Littlewood circle method in analytic number theory, and the use of adele rings, making this one of the unifying principles across number theory.
Read more about this topic: Local Analysis
Famous quotes containing the words number and/or theory:
“He is the richest man who knows how to draw a benefit from the labors of the greatest number of men, of men in distant countries, and in past times.”
—Ralph Waldo Emerson (1803–1882)
“The theory of the Communists may be summed up in the single sentence: Abolition of private property.”
—Karl Marx (1818–1883)