The question is in the title, and I do not really have anything to add. Nevertheless I had to write something here in order to be able to ask the question. Thanks.



Is there always, for a given prime $p$, a prime $\ell<p$ that is not a quadratic residue mod $p$?
Of course. Take a quadratic nonresidue $1\leq n\leq p-1$, then some prime divisor $\ell$ of $n$ will be a quadratic nonresidue.


See [this MO question](https://mathoverflow.net/questions/52211/effective-chebotarev-density/52237#52237) for what is known about number fields.


