Conjugate reducibility of families of block -diagonal matrices over an extension field of a perfect field, and applications to matrix subalgebras and subgroups.
Item
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Title
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Conjugate reducibility of families of block -diagonal matrices over an extension field of a perfect field, and applications to matrix subalgebras and subgroups.
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Identifier
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AAI3115232
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identifier
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3115232
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Creator
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Brock, Martin L.
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Contributor
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Adviser: Martin Moskowitz
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Date
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2004
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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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For L/k a field extension, k perfect, and M(n, F) the n x n matrices over field F, the main question is when can certain families of M( n, L) be conjugated into M(n, k), by an operator from Gℓ(n, L). We say such a family is k-rationalizable over L. Henceforth, let L be any extension of k¯. We also determine the maximal subsets of M(n, L) that can normalize a diagonal family of M(n, L) and be k-rationalized over L together with it. The primary methods are Galois Theory and the imbeddings of a field extension into the algebraic closure of its ground field.;Chapter 1 introduces a set of matrices viewable as block-diagonal "discriminant" matrices, and another viewable as block-horizontal "discriminant" matrices.;Chapter 2 exhibits all invertible matrices, that k-rationalize over k¯, certain block-diagonal "discriminant" subsets. Those emerge, basically, as the block-horizontal "discriminant" matrices. Results over k¯ are then extended to any extension of k¯.;Chapter 3 exhibits general block-diagonal subsets, with suitably restricted centralizers, which are k-rationalized over L by some block-horizontal "discriminant" matrix. Those emerge as the block-diagonal "discriminant" matrices; thus a reverse form of the previous.;Chapter 4 exhibits all diagonal families of M( n, L) which are k-rationalizable over L , and an important tight property of this exhibition. Chapter 5 gives an alternative, aesthetic, and quick way to "see" them.;Chapter 6 exhibits, for any k-rationalizable over L diagonal family of M(n, L), a superset for its normalizing elements in Gℓ( n, L) which can be k-rationalized over L together with the family. This superset is achieved with certain "maximal" diagonal families. Exhibition results are given for certain non-"maximal" diagonal families, and even block-diagonal families.;Chapter 7 gives applications to the diagonalizable subsets of M(n, k), k a perfect field. These include exhibitions and classifications of the subsets of M( n, k) that are: diagonalizable, second centralizers of some diagonalizable subset, and maximal diagonalizable; with their centralizers, second centralizers, and normalizers (or strong inclusions/imbeddings thereof) in M( n, k) and M(n, L).;Applications to M(n, k) can easily be a research subject: finite solvable or torsional abelian subgroups; k = Q or GF(pn); small n; the solvable subgroups of M(n, Z2 ).
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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.
