Current Search:  Algebraic number theory (x)

View All Items

  • CSV Spreadsheet
(1 - 6 of 6)
CHARACTERIZATION OF PRODUCT FORMULA FIELDS
Title
A CHARACTERIZATION OF PRODUCT FORMULA FIELDS.
Creator
HELLMAN, ALLEN PAUL, Florida Atlantic University
Abstract/Description
In this thesis we present a characterization of fields which admit a product formula. We prove that a field which admits a product formula consisting of admissible prime spots is a global field. This result was originally proved by Artin and Whaples in 1945. By limiting the admissible prime spots to those that are archimedean or discrete with finite residue class field, we are able to obtain a more elementary proof than that given by Artin and Whaples. The proof given here is, to our...
Show moreIn this thesis we present a characterization of fields which admit a product formula. We prove that a field which admits a product formula consisting of admissible prime spots is a global field. This result was originally proved by Artin and Whaples in 1945. By limiting the admissible prime spots to those that are archimedean or discrete with finite residue class field, we are able to obtain a more elementary proof than that given by Artin and Whaples. The proof given here is, to our knowledge, The render should notice that Artin and Whaples obtain, as a part of their result, that only the two types of prime spots mentioned above can occur in a product formula.
Show less
Date Issued
1973
PURL
http://purl.flvc.org/fcla/dt/13582
Subject Headings
Algebraic fields, Algebraic number theory
Format
Document (PDF)
Unique decomposition of direct sums of ideals
Title
Unique decomposition of direct sums of ideals.
Creator
Ay, Basak., Charles E. Schmidt College of Science, Department of Mathematical Sciences
Abstract/Description
We say that a commutative ring R has the unique decomposition into ideals (UDI) property if, for any R-module which decomposes into a finite direct sum of indecomposable ideals, this decomposition is unique up to the order and isomorphism class of the ideals. In a 2001 paper, Goeters and Olberding characterize the UDI property for Noetherian integral domains. In Chapters 1-3 the UDI property for reduced Noetherian rings is characterized. In Chapter 4 it is shown that overrings of one...
Show moreWe say that a commutative ring R has the unique decomposition into ideals (UDI) property if, for any R-module which decomposes into a finite direct sum of indecomposable ideals, this decomposition is unique up to the order and isomorphism class of the ideals. In a 2001 paper, Goeters and Olberding characterize the UDI property for Noetherian integral domains. In Chapters 1-3 the UDI property for reduced Noetherian rings is characterized. In Chapter 4 it is shown that overrings of one-dimensional reduced commutative Noetherian rings with the UDI property have the UDI property, also. In Chapter 5 we show that the UDI property implies the Krull-Schmidt property for direct sums of torsion-free rank one modules for a reduced local commutative Noetherian one-dimensional ring R.
Show less
Date Issued
2010
PURL
http://purl.flvc.org/FAU/2683133
Subject Headings
Algebraic number theory, Modules (Algebra), Noetherian rings, Commutative rings, Algebra, Abstract
Format
Document (PDF)
Minimal zero-dimensional extensions
Title
Minimal zero-dimensional extensions.
Creator
Chiorescu, Marcela, Florida Atlantic University, Charles E. Schmidt College of Science, Department of Mathematical Sciences
Abstract/Description
The structure of minimal zero-dimensional extension of rings with Noetherian spectrum in which zero is a primary ideal and with at most one prime ideal of height greater than one is determined. These rings include K[[X,T]] where K is a field and Dedenkind domains, but need not be Noetherian nor integrally closed. We show that for such a ring R there is a one-to-one correspondence between isomorphism classes of minimal zero-dimensional extensions of R and sets M, where the elements of M are...
Show moreThe structure of minimal zero-dimensional extension of rings with Noetherian spectrum in which zero is a primary ideal and with at most one prime ideal of height greater than one is determined. These rings include K[[X,T]] where K is a field and Dedenkind domains, but need not be Noetherian nor integrally closed. We show that for such a ring R there is a one-to-one correspondence between isomorphism classes of minimal zero-dimensional extensions of R and sets M, where the elements of M are ideals of R primary for distinct prime ideals of height greater than zero. A subsidiary result is the classification of minimal zero-dimensional extensions of general ZPI-rings.
Show less
Date Issued
2009
PURL
http://purl.flvc.org/FAU/210447
Subject Headings
Algebra, Abstract, Noetherian rings, Commutative rings, Modules (Algebra), Algebraic number theory
Format
Document (PDF)
study of divisors and algebras on a double cover of the affine plane
Title
A study of divisors and algebras on a double cover of the affine plane.
Creator
Bulj, Djordje., Charles E. Schmidt College of Science, Department of Mathematical Sciences
Abstract/Description
An algebraic surface defined by an equation of the form z2 = (x+a1y) ... (x + any) (x - 1) is studied, from both an algebraic and geometric point of view. It is shown that the surface is rational and contains a singular point which is nonrational. The class group of Weil divisors is computed and the Brauer group of Azumaya algebras is studied. Viewing the surface as a cyclic cover of the affine plane, all of the terms in the cohomology sequence of Chase, Harrison and Roseberg are computed.
Date Issued
2012
PURL
http://purl.flvc.org/FAU/3355618
Subject Headings
Algebraic number theory, Geometry, Data processing, Noncommutative differential geometry, Mathematical physics, Curves, Algebraic, Commutative rings
Format
Document (PDF)
existence of minimal logarithmic signatures for classical groups
Title
The existence of minimal logarithmic signatures for classical groups.
Creator
Singhi, Nikhil., Charles E. Schmidt College of Science, Department of Mathematical Sciences
Abstract/Description
A logarithmic signature (LS) for a nite group G is an ordered tuple = [A1;A2; : : : ;An] of subsets Ai of G, such that every element g 2 G can be expressed uniquely as a product g = a1a2 : : : ; an, where ai 2 Ai. Logarithmic signatures were dened by Magliveras in the late 1970's for arbitrary nite groups in the context of cryptography. They were also studied for abelian groups by Hajos in the 1930's. The length of an LS is defined to be `() = Pn i=1 jAij. It can be easily seen that for a...
Show moreA logarithmic signature (LS) for a nite group G is an ordered tuple = [A1;A2; : : : ;An] of subsets Ai of G, such that every element g 2 G can be expressed uniquely as a product g = a1a2 : : : ; an, where ai 2 Ai. Logarithmic signatures were dened by Magliveras in the late 1970's for arbitrary nite groups in the context of cryptography. They were also studied for abelian groups by Hajos in the 1930's. The length of an LS is defined to be `() = Pn i=1 jAij. It can be easily seen that for a group G of order Qk j=1 pj mj , the length of any LS for G satises `() Pk j=1mjpj . An LS for which this lower bound is achieved is called a minimal logarithmic signature (MLS). The MLS conjecture states that every finite simple group has an MLS. If the conjecture is true then every finite group will have an MLS. The conjecture was shown to be true by a number of researchers for a few classes of finite simple groups. However, the problem is still wide open. This dissertation addresses the MLS conjecture for the classical simple groups. In particular, it is shown that MLS's exist for the symplectic groups Sp2n(q), the orthogonal groups O 2n(q0) and the corresponding simple groups PSp2n(q) and 2n(q0) for all n 2 N, prime power q and even prime power q0. The existence of an MLS is also shown for all unitary groups GUn(q) for all odd n and q = 2s under the assumption that an MLS exists for GUn 1(q). The methods used are very general and algorithmic in nature and may be useful for studying all nite simple groups of Lie type and possibly also the sporadic groups. The blocks of logarithmic signatures constructed in this dissertation have cyclic structure and provide a sort of cyclic decomposition for these classical groups.
Show less
Date Issued
2011
PURL
http://purl.flvc.org/FAU/3172943
Subject Headings
Finite groups, Abelian groups, Number theory, Combinatorial group theory, Mathematical recreations, Linear algebraic groups, Lie groups
Format
Document (PDF)
the minimal logarithmic signature conjecture
Title
On the minimal logarithmic signature conjecture.
Creator
Singhi, Nidhi., Charles E. Schmidt College of Science, Department of Mathematical Sciences
Abstract/Description
The minimal logarithmic signature conjecture states that in any finite simple group there are subsets Ai, 1 i s such that the size jAij of each Ai is a prime or 4 and each element of the group has a unique expression as a product Qs i=1 ai of elements ai 2 Ai. Logarithmic signatures have been used in the construction of several cryptographic primitives since the late 1970's [3, 15, 17, 19, 16]. The conjecture is shown to be true for various families of simple groups including cyclic groups,...
Show moreThe minimal logarithmic signature conjecture states that in any finite simple group there are subsets Ai, 1 i s such that the size jAij of each Ai is a prime or 4 and each element of the group has a unique expression as a product Qs i=1 ai of elements ai 2 Ai. Logarithmic signatures have been used in the construction of several cryptographic primitives since the late 1970's [3, 15, 17, 19, 16]. The conjecture is shown to be true for various families of simple groups including cyclic groups, An, PSLn(q) when gcd(n; q 1) is 1, 4 or a prime and several sporadic groups [10, 9, 12, 14, 18]. This dissertation is devoted to proving that the conjecture is true for a large class of simple groups of Lie type called classical groups. The methods developed use the structure of these groups as isometry groups of bilinear or quadratic forms. A large part of the construction is also based on the Bruhat and Levi decompositions of parabolic subgroups of these groups. In this dissertation the conjecture is shown to be true for the following families of simple groups: the projective special linear groups PSLn(q), the projective symplectic groups PSp2n(q) for all n and q a prime power, and the projective orthogonal groups of positive type + 2n(q) for all n and q an even prime power. During the process, the existence of minimal logarithmic signatures (MLS's) is also proven for the linear groups: GLn(q), PGLn(q), SLn(q), the symplectic groups: Sp2n(q) for all n and q a prime power, and for the orthogonal groups of plus type O+ 2n(q) for all n and q an even prime power. The constructions in most of these cases provide cyclic MLS's. Using the relationship between nite groups of Lie type and groups with a split BN-pair, it is also shown that every nite group of Lie type can be expressed as a disjoint union of sets, each of which has an MLS.
Show less
Date Issued
2011
PURL
http://purl.flvc.org/FAU/3172946
Subject Headings
Finite groups, Abelian groups, Number theory, Combinatorial group theory, Mathematical recreations, Linear algebraic groups, Lie groups
Format
Document (PDF)