(Both the published version and the arXiv version are available online.)

We say that a polynomial f(x) with coefficients in the ring R is decomposable (over R) if we can write f as the composition of two polynomials over R which have strictly smaller degrees than f. We say that f is indecomposable (over R) if it has degree at least two and is not decomposable. In this note we answer two questions posed in a recent paper by Gusić:

Question 1: Prove or disprove. Let R be an integral domain of zero characteristic. Let S denote the integral closure of R in the field of fractions of R. Assume that S≠R. Then there exists a monic polynomial f over R that is decomposable over S but not over R.

Question 2: Prove or disprove. Let R be the ring of integers of a number field K. Assume that R is not a unique factorization domain. Then there exists a polynomial f over R that is decomposable over K but not over R.

We show that the first question has a negative answer, and the second question has a positive answer.

These questions were motivated by two results due to Turnwald, which assert that if R is an integral domain of characteristic zero, and K is a field containing R, then
(1) If R is integrally closed in its field of fractions, then every indecomposable monic polynomial over R is indecomposable over K.
(2) If R is a unique factorization domain, then every indecomposable polynomial over R is indecomposable over K.

Michael Zieve:  home page   publication list