Hasse principle

Builds on p-adic number Diophantine equation

Definition

For a prime $p$ , let $\Q_p$ be the field of p-adic numbers, and let $\Q_\infty = \R$ . Let $\mathcal{C}$ be a class of systems of polynomial equations with coefficients in $\Q$ . For a system of homogeneous equations, a solution means a solution other than the zero solution. Let $V$ be the set of all primes together with the symbol $\infty$ . The class $\mathcal{C}$ satisfies the Hasse principle if each system in $\mathcal{C}$ with a solution in $\Q_v$ for every $v \in V$ has a solution in $\Q$ .

Remarks

The converse always holds, since $\Q$ is a subfield of each $\Q_v$ . The same definition makes sense over any number field, with its completions in place of the $\Q_v$ .

Example: the class of homogeneous quadratic equations $Q(x_1, \dots, x_n) = 0$ , with $Q$ a quadratic form over $\Q$ , satisfies the Hasse principle; this is the Hasse–Minkowski theorem.

Non-example: the class of homogeneous cubic equations in three variables does not. Selmer's equation $3x^3 + 4y^3 + 5z^3 = 0$ has nonzero solutions in $\R$ and in every $\Q_p$ , but it has no nonzero solution in $\Q$ .

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Tag	`SP8H3S2XWX`
URI	`m2://SP8H3S2XWX`
Kind	definition
Writing	W0 stub
MSC	11D09, 11E12, 11G05
Defines	Hasse principle
Uses	p-adic number, Diophantine equation
Changed	2 November 2026

Hasse principle

Definition

Remarks

Reading path

Further reading

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