Let $q$ be a prime power. I will use the notations of Keith Conrad's *Carlitz extensions* paper (but I'll work over $\mathbb{F}_q$ rather than $\mathbb{F}_p$).

The most general question I'm asking here is whether there is anything resembling the classical theory of quadratic residues (including results like quadratic reciprocity, supplementary laws, Gauss sums, Gauss' lemma etc.) in the context of Carlitz polynomials -- i. e., over $\mathbb{F}_q\left[T\right]$ instead of $\mathbb Z$, with "squaring" replaced by application of $\left[N\right]$ for some fixed polynomial $N\in\mathbb{F}_q\left[T\right]$. (This is in line with the usual idea that $\mathbb{F}_q\left[T\right]$ is an analogue of $\mathbb Z$, and applying a Carlitz polynomial in the former ring is like taking a power in the latter.)

Of course, unlike $\mathbb Z$, where every prime apart from $2$ is $\equiv 1 \mod 2$, there is no irreducible $N\in\mathbb{F}_q\left[T\right]$ such that all but finitely many monic irreducibles in $\mathbb{F}_q\left[T\right]$ are $\equiv 1\mod N$. So there is no obvious analogue of squaring that would mirror the "the product of two nonsquare residues is a square residue" property. But there is still an irreducible in $\mathbb{F}_q\left[T\right]$ which, in some sense, is simpler than the others: namely, $T$. Whenever $\pi\in\mathbb{F}_q\left[T\right]$ is an irreducible such that $\pi\equiv 1\mod T$, the elements $u$ of $\mathbb{F}_q\left[T\right] / \pi$ which can be written as $\left[T\right]\left(v\right)=v^q+Tv$ for $v\in\mathbb{F}_q\left[T\right] / \pi$ form an $\mathbb{F}_q\left[T\right]$-submodule of the Carlitz module $C\left(\mathbb{F}_q\left[T\right] / \pi\right)$. They can be viewed as the elements annihilated by $\left[\dfrac{\pi-1}{T}\right]$.

My more concrete question is in how far they share properties with quadratic residues in $\mathbb Z$. I don't dare formulate any conjectures (lacking computational data and number-theoretical intuition), but one could ask how the property of a monic irreducible $\phi$ to be a "$T$-quadratic residue" modulo another $\psi$ correlates with the same in the other direction.

(Fun fact: From the cyclicity of the Carlitz module $C\left(\mathbb{F}_q\left[T\right] / \pi\right)$, it follows immediately that if $\pi$ is a monic irreducible in $\mathbb{F}_q\left[T\right]$ satisfying $\pi\equiv 1\mod T$, then $-T$ is a $p-1$-th power in $\mathbb{F}_q\left[T\right]$. Of course, this also follows from Hilbert's theorem 90 or cyclicity of the group $\left(\mathbb{F}_q\left[T\right] / \pi\right)^\times$.)

If $T$ is a bad (because not invariant under $\mathbb{F}_q$-automorphisms or for whatever other reason) analogue of $2$, we might consider $T^q - T$ instead -- it doesn't appear like irreducibility is important here...