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Title
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RELATIVE INTEGRAL BASIS IN ALGEBRAIC NUMBER FIELDS.
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Identifier
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AAI8222948
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identifier
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8222948
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Creator
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HAGHIGHI, MAHMOOD.
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Contributor
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Charles P. Smith | Samuel Messick
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Date
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1982
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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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Part 1 is devoted to preliminary definitions and theorems, that are crucial for the rest of this work. In part 2, as is well known, in {lcub}K:k{rcub} = n for O(,k) = P.I.D., the O(,K) is a free O(,k)-module. So we have computed a free-basis of.;(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI).;for (d(,K(,1)),d(,K(,2))) = 1 and (NOT=) 1 in four cases depending on m(,i). If O(,k) (NOT=) P.I.D., then a free basis of O(,K)/O(,k) may or may not exist as shown directly. For the first time Speiser and Hecke gave a criterion when {lcub}K:k{rcub} = 2. From that we show.;(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI).;has a free basis if 3 (VBAR) h(,3) in type I. If 3 (VBAR) h(,3) then for special huge values of n we show a free basis can exist. In part 3, by using properties of finitely generated modules over Dedekind domains, we show.;(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI).;in {lcub}K:k{rcub} = n. From this O(,K)/O(,k) has a free basis () I = P.I. in O(,k), but the construction of I for n > 2 is difficult as we have shown. So from.;(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI).;and a property of D(,K/k), we show I('2)(TURN)d(,K/k). From this for h(,k) = odd, O(,K)/O(,k) has free basis () d(,K/k) = P.I. As a corollary for O(,k) = P.I.D., a free basis exists for all {lcub}K:k{rcub} = n. In part 4 we compute I for O(,K(,4)) (TURNEQ) O(,K(,1))(CRPLUS) I for (d(,K(,1)),d(,K(,2))) = 1 and (NOT=)1, that shows for I (TURNEQ) O(,K(,1)) a free basis exists, otherwise not. We also compute a free basis for {lcub}K:k{rcub} = 2, and since O(,6)/O(,3) has free basis () 3(,1) = P.I., then for it we also compute a free basis for 3(,1) = P.I. Finally since.;(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI).;so O(,6)/O(,3) has free basis for any value of n. We compute a free basis for 3(,1) = P.I. for Type 1 and Type 2, and also we compute a free basis for all n = 3t + 1, -n (square-free), no matter whether 3(,1) = P.I. or not.
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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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Psychology
