in Proc. of 7th Int'l Conf. on Fibonacci Numbers (Graz 1996) G.E. Bergum et al eds, Kluwer 1998.

Binomial Coefficients Generalized with respect to a Discrete Valuation

Abstract: By a theorem of Kummer, the exact power of p (prime) dividing the binomial coefficient "n choose k" equals the number of carries that occur when k is added to n-k in base p arithmetic. We show an analogue of this theorem for certain generalized binomial coefficients that arise naturally in the study of integer-valued polynomials. (A polynomial f with coefficients in the quotient field K of a domain D is called integer-valued, if for all d in D, f(d) is again in D.)

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.

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