Finite fields are indispensable to modern cryptography and coding theory. It is well known that the implementation of the arithmetic of finite fields greatly depends on their representation, which in turn depends on the existence of irreducible polynomials of special forms. So, studying irreducible polynomials over finite fields besides having a natural theoretical appeal, has an applied appeal, too. There are an abundant number of known results and open problems about the distribution of irreducible polynomials over finite fields. In this talk, we briefly survey what is known and what remains to be explored.

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