There are many results about irreducible polynomials over finite fields:

we know a cardinality of all irreducible polynomials with given degree, we know explicit examples of irreducible polynomials, we know effective algorithms for construction irreducible polynomials, and for verifying that given polynomial is irreducible.

Are there similar results about irreducible algebraic sets? Could you give a reference?

Thank you!

UPD: by explicit examples of irreducible polynomials I mean the following: $x^p - x + a$ is irreducible over $\mathbb{F}_q$ if $a \not= 0$. Are there are similar examples for an arbitrary dimension?

I want to have such explicit examples of irreducible algebraic sets because I hope it can help to solve my another question: an algebraic variety for a boolean circuit