The Eisenstein criterion is one of few simple sufficient criteria for determining the irreducibility of rational coefficient polynomials. The last 15 years has yielded results regarding the counting of irreducible polynomials of bounded height. This will encompass the Eisenstein criterion and two generalisations of the criterion. Some of the results are joint work with Igor Shparlinski.
Pairwise coprimality is an essential aspect of the Chinese remainder theorem which is used in areas such as encryption.
I will present some recent results about the counting of arrays of integers that satisfy certain pairwise coprimality conditions. One of the results is joint work with Juan Arias de Reyna.