Hasse principle
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In mathematics, Helmut Hasse's local-global principle, also known as the Hasse principle, is the assertion that an equation can be solved over the rational numbers if and only if it can be solved over the real numbers and over the p-adic numbers for every prime p.
The Hasse-Minkowski theorem states that the local-global principle holds for quadratic forms over the rational numbers (which is Minkowski's result); and more generally over any number field (as proved by Hasse), when one uses all the appropriate local field necessary conditions. Hasse's theorem on cyclic extensions states that the local-global principle applies to the condition of being a relative norm for a cyclic extension of number fields.
For algebraic forms of higher degree d the situation is more complicated; basically, if the number of variables is n, one expects the principle to hold when n ≥ N(d). For example N(3) can be taken to be 9 (research is ongoing in this area); but cannot be very small, since the Hasse principle does fail for cubics in a small number of variables (for example, elliptic curves, corresponding to n = 3). The 'large number of variables' results depend on the Hardy-Littlewood circle method, which was extended to all number fields by C. L. Siegel (quadratic case) and C. P. Ramanujam (in general). According to an idea of Manin, the obstructions to the Hasse principle holding for cubic forms can be tied into the theory of the Brauer group; it is only recently that it has been shown that this setting isn't the complete story (Alexei Skorobogatov, 1999).
See also
External links
- PlanetMath article (http://planetmath.org/encyclopedia/HassePrinciple.html)
- Swinnerton-Dyer, Diophantine Equations: Progress and Problems, online notes (http://swc.math.arizona.edu/notes/files/DLSSw-Dyer1.pdf)