Arithmetic in the Ring of Formal Power Series with Integer Coefficients

The divisibility and factorization theory of the integers and of the ring of polynomials (in one variable) over the integers are standard topics in a first course in abstract algebra. Concepts such as prime, irreducible, and invertible elements, unique factorization, and irreducibility criteria are extensively studied and are part of the core of the course. On the other hand, the natural extension of the ring of polynomials R[x] over R, namely the ring R[[x]] of formal powers series in one variable over R, is hardly ever mentioned in such a course. In most cases, it is relegated to the homework problems (or to the exercises in the textbooks), and one learns that, like R[x], R[[x]] is an integral domain provided that R is an integral domain. More surprising is to learn that, in contrast to the situation of polynomials, in R[[x]] there are many invertible elements: while the only units in R[x] are the units of R, a necessary and sufficient condition for a power series to be invertible is that its constant term be invertible in R. This fact makes the study of arithmetic in R[[x]] simple when R is a field: the only prime element is the variable x. As might be expected, the study of prime factorization in Z[[x]] is much more interesting (and complicated), but to the best of our knowledge it is not treated in detail in the available literature. After some basic considerations, it is apparent that the question of deciding whether or not an integral power series is prime is a difficult one, and it seems worthwhile to develop criteria to determine irreducibility in Z[[x]] similar to Eisenstein’s criterion for polynomials. In this note we propose an easy argument that provides us with an infinite class of irreducible power series over Z. As in the case of Eisenstein’s criterion in Z[x], our criteria give only sufficient conditions, and the question of whether or not a given power series is irreducible remains open in a vast array of cases, including quadratic polynomials. It is important to note that irreducibility in Z[x] and in Z[[x]] are, in general, unrelated. For instance, 6 + x+ x is irreducible in Z[x] but can be factored in Z[[x]], while 2 + 7x+ 3x is irreducible in Z[[x]] but equals (2 + x)(1 + 3x) as a polynomial (observe that this is not a proper factorization in Z[[x]] since 1+ 3x is invertible). For some particular quadratic polynomials we are able to answer the question of irreducibility, but we leave (among many others) some questions for further research: Is there a definite criterion to decide whether a quadratic