Bhargava factorials and irreducibility of integer-valued polynomials. (English) Zbl 1493.13028

Prasad, Devendra

Rocky Mt. J. Math. 52, No. 3, 1031-1038 (2022).

For a set \(S\) of rational integers the author considers the ring Int\((S,Z)\) of polynomials \(f\in Q[X]\) satisfying \(f(S)\subset Z\). For given integers \(d,k\) the author introduces \(d_k\)-orderings in the following way:
For every prime divisor \(p\) of \(d\) let \(\{u_p(i)\}\) be a \(p\)-ordering of \(S\), as defined by A. M. S. Ramasamy and S. P. Mohanty [J. Indian Math. Soc., New Ser. 62, No. 1–4, 210–214 (1996; Zbl 0899.11011)]. A \(d_k\)-ordering of \(S\) is a sequence \(x_1,x_2,\dots,x_k\) of elements of \(S\) satisfying \[ x_i\equiv u_p(i)\pmod{p^{e_k+1}}\ (1\le i\le k) \] for all \(p|d\), with \(p^{e_k}\) being the largest power of \(p\) dividing \[ \prod_{q|d\ prime}\prod_{i=0}^{k-1}\left(u_q(k)-u_q(i)\right). \] This notion is used to provide a criterion of irreducibility for polynomials \(f\in\mathrm{Int}(S,Z)\) (Theorem 3.6). The author states at the end of the paper that his result can be carried over to arbitrary Dedekind domains.

Reviewer: Władysław Narkiewicz (Wrocław)

MSC:

13F20	Polynomial rings and ideals; rings of integer-valued polynomials
11B83	Special sequences and polynomials
11C08	Polynomials in number theory
13B25	Polynomials over commutative rings
13F05	Dedekind, Prüfer, Krull and Mori rings and their generalizations

Keywords:

integer-valued polynomials; irreducible elements; generalized factorial

Citations:

Zbl 0899.11011

References:

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