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](https://mathoverflow.net/questions/237164/an-algebraic-variety-for-a-boolean-circuit)



Irreducible algebraic sets via irreducible polynomials
Bjorn Poonen and his coauthors have studied topics of this type in a series of papers on "Bertini Theorem over Finite Fields".


See for example the paper:


Poonen, Charles - Bertini irreducibility theorems over finite fields.


<http://www-math.mit.edu/~poonen/papers/bertini_irred.pdf>


Edit: I should clarify; this answer addresses the first part of the question, which asks about generalisations of "cardinality of all irreducible polynomials with given degree".


