# Infinitely generated symbolic Rees algebras over finite fields

## TL;DR

This paper demonstrates that for a polynomial ring with twelve variables over any field, there exists a prime ideal whose symbolic Rees algebra is not finitely generated, highlighting limitations in algebraic structure.

## Contribution

It provides the first example of a prime ideal with a non-finitely generated symbolic Rees algebra in a polynomial ring over an arbitrary field.

## Key findings

- Existence of prime ideal with non-finitely generated symbolic Rees algebra
- Applicable to polynomial rings over any field
- Highlights limitations in algebraic finiteness properties

## Abstract

For the polynomial ring over an arbitrary field with twelve variables, there exists a prime ideal whose symbolic Rees algebra is not finitely generated.