TITLE: applications of finding least quadratic nonresidue mod $p$?
QUESTION [5 upvotes]: 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.

REPLY [2 votes]: Let me expand my comment into a short answer. 
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 $$\mathbb{F}_{p^2}=\mathbb{F}_p[x]/(x^2-a).$$