In mathematics, Helmut Hasse's local–global principle, also known as the Hasse principle, is the idea that one can find an integer solution to an equation by using the Chinese remainder theorem to piece together solutions modulo powers of each different prime number. This is handled by examining the equation in the completions of the rational numbers: the real numbers and the p-adic numbers. A more formal version of the Hasse principle states that certain types of equations have a rational solution if and only if they have a solution in the real numbers and in the p-adic numbers for each prime p.
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, or number fields. For number fields, rather than reals and p-adics, one uses complex embeddings and -adics, for prime ideals .
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 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.
A counterexample by Ernst S. Selmer shows that the Hasse–Minkowski theorem cannot be extended to forms of degree 3: The cubic equation 3x3 + 4y3 + 5z3 = 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.
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.
This course is a modern exposition of "Duke's Theorems" which describe the distribution of representations of large integers by a fixed ternary quadratic form. It will be the occasion to introduce the
In mathematics, an algebraic number field (or simply number field) is an extension field of the field of rational numbers such that the field extension has finite degree (and hence is an algebraic field extension). Thus is a field that contains and has finite dimension when considered as a vector space over . The study of algebraic number fields, and, more generally, of algebraic extensions of the field of rational numbers, is the central topic of algebraic number theory.
In mathematics, a rational number is a number that can be expressed as the quotient or fraction \tfrac p q of two integers, a numerator p and a non-zero denominator q. For example, \tfrac{-3}{7} is a rational number, as is every integer (e.g., 5 = 5/1). The set of all rational numbers, also referred to as "the rationals", the field of rationals or the field of rational numbers is usually denoted by boldface Q, or blackboard bold \Q. A rational number is a real number.
In mathematics, Galois cohomology is the study of the group cohomology of Galois modules, that is, the application of homological algebra to modules for Galois groups. A Galois group G associated to a field extension L/K acts in a natural way on some abelian groups, for example those constructed directly from L, but also through other Galois representations that may be derived by more abstract means. Galois cohomology accounts for the way in which taking Galois-invariant elements fails to be an exact functor.
Explores integral Hasse principle, locally isometric lattices, and adèles rings.
Explores integral representations, quadratic lattices, and the Hasse Principle.
Explores primes in arithmetic progression, focusing on L-functions, characters, and the divergence of the sum of 1 over p for p congruent to a modulo q.
We establish p-adic versions of the Manin-Mumford conjecture, which states that an irreducible subvariety of an abelian variety with dense torsion has to be the translate of a subgroup by a torsion point. We do so in the context of certain rigid analytic s ...
We answer several questions posed by Beck, Cox, Delgado, Gubeladze, Haase, Hibi, Higashitani, and Maclagan in [Cox et al. 14, Question 3.5 (1),(2), Question 3.6], [Beck et al. 15, Conjecture 3.5(a),(b)], and [Hasse et al. 07, Open question 3 (a),(b) p. 231 ...
A classical result of Hasse states that the norm principle holds for finite cyclic extensions of global fields, in other words local norms are global norms. We investigate the norm principle for finite dimensional commutative kale algebras over global fiel ...