How to find quadratic residues in the polynomial ring $k[t]$?

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I have a question:

Given a field $k$, finite or infinite, and an element $p(t)$ in the polynomial ring $k[t]$.

I am searching for results of any kind about how to find quadratic residues in the quotient ring $k[t]/⟨p(t)⟩$, if they exist. Can anyone recommend a good book/paper on that topic?

(There is an answer in: Literature on Quadratic Residues in Polynomial Quotient ring for this same question with a link that does not work.)