##  [Bad Reduction](/bad-reduction) 

  ##  [Bad Reduction](https://puremath.quantumdictionary.io/bad-reduction-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 prime (or place) of a number field at which an algebraic variety, scheme, or model acquires singularities or otherwise degenerates after reduction modulo that prime, so the special fiber is not smooth and arithmetic properties that rely on smoothness may fail.

 

 

 

 

 





 

 



 ##  [Bad Reduction](https://algebra.quantumdictionary.io/bad-reduction-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 place (a closed point of the base, a prime of a number ring, or a special parameter) in an arithmetic or geometric family where the special fiber fails to be smooth or acquires extra singular components; informally, the family has 'bad' behavior at that place compared with 'good reduction' where fibers are smooth and well behaved.