کد مقاله | کد نشریه | سال انتشار | مقاله انگلیسی | نسخه تمام متن |
---|---|---|---|---|
6415596 | 1630677 | 2014 | 15 صفحه PDF | دانلود رایگان |
Let K be a number field of degree n with ring of integers OK. By means of a criterion of Gilmer for polynomially dense subsets of the ring of integers of a number field, we show that, if hâK[X] maps every element of OK of degree n to an algebraic integer, then h(X) is integral-valued over OK, that is, h(OK)âOK. A similar property holds if we consider the set of all algebraic integers of degree n and a polynomial fâQ[X]: if f(α) is integral over Z for every algebraic integer α of degree n, then f(β) is integral over Z for every algebraic integer β of degree smaller than n. This second result is established by proving that the integral closure of the ring of polynomials in Q[X] which are integer-valued over the set of matrices Mn(Z) is equal to the ring of integral-valued polynomials over the set of algebraic integers of degree equal to n.
Journal: Journal of Number Theory - Volume 137, April 2014, Pages 241-255