Conjugate reducibility of families of block-diagonal matrices over an extension field of a perfect field, and applications to matrix subalgebras and subgroups

For L/k a field extension, k perfect, and M(n, F) the n × 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,$Z2$).