In the context of constructing integer-valued polynomials as linear combinations of generalized binomial polynomials, we are also led to construct integral bases for the ring of algebraic integers in a number field with special properties with respect to a prime ideal P.
For instance, if the prime p splits in R as pR=P^e (with [R:P]=p^f) then we construct a Z-basis of R with the property that (for k ranging from 0 to n) r is in P^k if and only if the first kf coefficients in the representation of r as Z-linear combination of the basis elements are divisible by p.
PDF: S. Frisch: Binomial coefficients generalized w.r.t. a discrete valuation
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