# applications of finding least quadratic nonresidue mod $p$?

I saw some papers from famous mathematicians (assuming GRH or without it) which are devoted to finding bound for least quadratic nonresidues modulo prime number $$p$$.

My question is that why it is so important and what are the applications of it?

Thanks.

• Is it a recent paper? What estimate have they proved? Anyway, this is a classical, notoriously difficult problem; so, definitely interesting on its own.
– Seva
Sep 3, 2019 at 12:38
• @Seva, no. they are classic. but I saw one related to from Terence Tao (The Elliott–Halberstam conjecture implies the Vinogradov least quadratic nonresidue conjecture).
• For instance, if we want to explicitly construct a field with $p^2$ elements, we can take the quotient $$\mathbb{F}_p[x]/(x^2-a),$$ where $a$ is not a square in $\mathbb{F}_p$. Sep 3, 2019 at 13:44
The explicit knowledge of a quadratic nonresidue $$a$$ modulo $$p$$ allows us to construct the irreducible quadratic polynomial $$x^2-a$$ in $$\mathbb{F}_p[x]$$, and so we have a very explicit description of the field with $$p^2$$ elements as $$\mathbb{F}_{p^2}=\mathbb{F}_p[x]/(x^2-a).$$