##  [Discriminant](/discriminant) 

  ##  [Discriminant](https://puremath.quantumdictionary.io/discriminant-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 scalar invariant associated to a polynomial, algebraic equation, or algebraic object (e.g., number field, bilinear form) that vanishes exactly when roots collide (multiple roots) and encodes arithmetic and ramification information; often computed as a determinant or as a product of squared differences of roots.

 

 

 

 

 





 

 



 ##  [Discriminant](https://algebra.quantumdictionary.io/discriminant-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 discriminant is a scalar invariant associated to a polynomial, algebraic form, or algebraic object that vanishes precisely when the object has singularities such as repeated roots, multiple factors, or non-transverse intersections.