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Dieudonne-Dwork Quotients

Dr. Shawn Clift

Dr. Shawn Clift / Eastern Kentucky University

Dieudonn-Dwork showed that T(x)^p/T(x^p)\in Z[[x]], with all positive power coefficients of this quotient series divisible by p. First we explore for what rings this holds. We then discuss the possibility of starting with U(x)\in Z[[x]] where U(0) = 1 and all the coefficients are divisible by a prime p and finding T(x) 2 Z[[x]] with T(0) = 1 such that T(x)^p/T(x^p) = U(x). We would also like to see if we can extend this idea to quotients of form T(x)^p^n/T(x^p^n) for n>=2 as well as taking a brief look at the topological groups formed by the special units of Z[[x]] .

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