##  [Brauer–Manin Obstruction](/brauer-manin-obstruction) 

  ##  [Brauer–Manin Obstruction](https://puremath.quantumdictionary.io/brauer-manin-obstruction-0) 

  

 [![Pure Mathematics Dictionary](/sites/default/files/styles/large/public/2026-01/Pure%20Mathematics.png.webp?itok=5pZnFQ59)](/topic-specific-dictionaries/mathematics-logic/pure-mathematics)

- Mathematics & Logic -

**Pure Mathematics Dictionary**

 







 

 

 

 



 

 

 

 

Definition

A mechanism detecting failure of the Hasse principle for rational points on varieties over global fields: elements of the Brauer group of a variety pair with adelic points via local invariants, and a nontrivial global obstruction (nonzero sum of local invariants) can rule out the existence of rational points even when local points exist everywhere.

 

 

 

 

 





 

 



 ##  [Brauer–Manin Obstruction](https://algebra.quantumdictionary.io/brauer-manin-obstruction-1) 

  

 [![Algebra](/sites/default/files/styles/large/public/2026-01/Algebra.png.webp?itok=3pHxBnUF)](/topic-specific-dictionaries/pure-mathematics/algebra)

- Pure Mathematics -

**Algebra Dictionary**

 







 

 

 

 



 

 

 

 

Definition

A mechanism using elements of the Brauer group of a variety to obstruct the existence of rational points even when local points exist everywhere; concretely, evaluation of Brauer classes at adelic points produces compatibility conditions whose failure explains counterexamples to the Hasse principle and refines weak approximation statements.