UNITS IN PARAMETERIZED P-ADATROPIC NUMBER FIELDS.
Item
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Title
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UNITS IN PARAMETERIZED P-ADATROPIC NUMBER FIELDS.
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Identifier
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AAI8023720
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identifier
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8023720
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Creator
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MAWYER, FARLEY.
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Contributor
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Harvey Cohn
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Date
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1980
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Language
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English
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Publisher
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City University of New York.
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Subject
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Mathematics
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Abstract
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Let p be a prime. Let f(x) = x('n) + a(,n-1) x('n-1) +...+ a(,1) x + a(,0),a(,i) (ELEM) Z, 0 (LESSTHEQ) i (LESSTHEQ) n. The polynomial f is p-adatropic if there are n + 1 consecutive integers, c(,i), such that (VBAR)f(c(,i))(VBAR) is a power of p for each i. This paper will attempt to expand on the work of H. Cohn on the calculation of units in fields generated by p-adatropic polynomials. The problem to be discussed here can be divided into three major parts. They are: (1) Find all p-adatropic polynomials of degree < 5 which are given in terms of one or more parameters. (2) Let f(x) be an irreducible p-adatropic polynomial and let f((THETA)) = 0. Let k = Q((THETA)). Using the fact that p-adatropic polynomials give many powers of p at values of x which differ by small integers, together with the fact that f(m) = N(,k/Q)(m-(THETA)), we factor the n + 1 consecutive ideals, ((THETA)-m(,i)), where (VBAR)f(m(,i))(VBAR) is a power of p. By looking at these ideals as elements of a vector space over Q, we are led to several units. (3) In the cases where we have r independent units, where r is the Dirichlet rank, we prove their independence by showing that the regulator is non-vanishing. This is done, using the computer as a guide, by finding bounds for the real roots of the parameterized defining polynomial and hence bounds for the conjugates of the units themselves.
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Type
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dissertation
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Source
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PQT Legacy CUNY.xlsx
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degree
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Ph.D.
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Program
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Mathematics
