Intuition
Given a polynomial equation with rational coefficients, if it has a rational solution, then this also yields a real solution and a ''p''-adic solution, as the rationals embed in the reals and ''p''-adics: a global solution yields local solutions at each prime. The Hasse principle asks when the reverse can be done, or rather, asks what the obstruction is: when can you patch together solutions over the reals and ''p''-adics to yield a solution over the rationals: when can local solutions be joined to form a global solution? One can ask this for other rings or fields: integers, for instance, orForms representing 0
Quadratic forms
The Hasse–Minkowski theorem states that the local–global principle holds for the problem of representing 0 by quadratic forms over the rational numbers (which is Minkowski's result); and more generally over anyCubic forms
A counterexample by Ernst S. Selmer shows that the Hasse–Minkowski theorem cannot be extended to forms of degree 3: The cubic equation 3''x''3 + 4''y''3 + 5''z''3 = 0 has a solution in real numbers, and in all p-adic fields, but it has no nontrivial solution in which ''x'', ''y'', and ''z'' are all rational numbers. Roger Heath-Brown showed that every cubic form over the integers in at least 14 variables represents 0, improving on earlier results of Davenport. Since every cubic form over the p-adic numbers with at least ten variables represents 0, the local–global principle holds trivially for cubic forms over the rationals in at least 14 variables. Restricting to non-singular forms, one can do better than this: Heath-Brown proved that every non-singular cubic form over the rational numbers in at least 10 variables represents 0, thus trivially establishing the Hasse principle for this class of forms. It is known that Heath-Brown's result is best possible in the sense that there exist non-singular cubic forms over the rationals in 9 variables that don't represent zero. However, Hooley showed that the Hasse principle holds for the representation of 0 by non-singular cubic forms over the rational numbers in at least nine variables. Davenport, Heath-Brown and Hooley all used theForms of higher degree
Counterexamples by Fujiwara and Sudo show that the Hasse–Minkowski theorem is not extensible to forms of degree 10''n'' + 5, where ''n'' is a non-negative integer. On the other hand, Birch's theorem shows that if ''d'' is any odd natural number, then there is a number ''N''(''d'') such that any form of degree ''d'' in more than ''N''(''d'') variables represents 0: the Hasse principle holds trivially.Albert–Brauer–Hasse–Noether theorem
The Albert–Brauer–Hasse–Noether theorem establishes a local–global principle for the splitting of a central simple algebra ''A'' over an algebraic number field ''K''. It states that if ''A'' splits over every completion ''K''''v'' then it is isomorphic to a matrix algebra over ''K''.Hasse principle for algebraic groups
The Hasse principle for algebraic groups states that if ''G'' is a simply-connected algebraic group defined over the global field ''k'' then the map from : is injective, where the product is over all places ''s'' of ''k''. The Hasse principle for orthogonal groups is closely related to the Hasse principle for the corresponding quadratic forms. and several others verified the Hasse principle by case-by-case proofs for each group. The last case was the group ''E''8 which was only completed by many years after the other cases. The Hasse principle for algebraic groups was used in the proofs of the Weil conjecture for Tamagawa numbers and the strong approximation theorem.See also
*Notes
References
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* {{springer, title=Hasse principle, id=p/h046670