Notation for Legendre symbol over polynomial rings

Question

Hopefully this is a simple question to answer. I am wondering if there is any standard notation for something equivalent to the Legendre symbol, but over polynomial rings. That is, if we have a prime $q$, is there a standard notation to indicate whether some $f \in \mathbb{F}_q[x]$ is a square or not?

Answer 1

Yes, we have an analogy of $\Bbb Z$ with $\Bbb F_p[X]$, and we can also look at quadratic reciprocity in $\Bbb F_p[X]$ via Euler's criterion. The details can be found in Lemmermayer, section $12.2$:

If $P$ is an irreducible polynomial over $\Bbb F_p$ with $p$ odd, and $NP=p^{{\rm deg}(P)}$, then $f^{NP−1}\equiv 1\bmod P$, hence $$ 0=f^{NP−1}-1=(f^{\frac{NP−1}{2}}-1)(f^{\frac{NP−1}{2}}+1) $$ Since $P$ is prime we conclude that $f^{\frac{NP−1}{2}}\equiv \pm 1 \bmod p$. Now define the Legendre symbol by $$ \left( \frac{f}{P}\right):=f^{\frac{NP−1}{2}}\bmod P. $$ And indeed we have $\left( \frac{f}{P}\right)=1$ if $f\equiv g^2\bmod P$ is a square modulo $P$.