Let m = m1f2 where m1 is a square-free positive integer and m is congruent to 1 or 2 mod 4. A theorem of Gauss (see [5]) states that the number of ways to write m as a sum of 3 squares is 12 times the size of the ring class group with discriminant -4m in the field \u211a(\u221a-m1). The proof given by Gauss involves the arithmetic of binary quadratic forms; Venkow (see [12]) obtained an alternative proof by embedding the field \u211a(\u221a-m1) in the quaternion algebra over \u211a. This thesis takes Venkow's proof as its starting point. We prove several further facts about the correspondence established by Venkow and apply these results to the study of imaginary quadratic ring class groups.
\r\n\r\nLet H denote the quaternion algebra over \u211a, let E denote the maximal order in H and let U denote the group of 24 units in E. Let B1(m) be the set of quaternions in E with trace 0 and norm m. The group U acts on B0(m) by conjugation; let B1(m) denote the set of orbits of B0(m) under the action of U. For \u00b5 = ui1 + vi2 + wi3\u03b5B1(m) we let [u,v,w] denote the orbit containing \u00b5.
\r\n\r\nVenkow proved Gauss's result by defining a sharply transitive action of \u0393(m), the ring class group with discriminant -4m, on B(m). In chapter 2 we establish some more subtle properties of this action. The prime 2 ramifies in the extension \u211a(\u221a-m1) and its prime divisor \u21192 is a regular ideal with respect to the discriminant -4m. It is shown that the class containing \u21192 maps [u,v,w] to [-u,-w,-v]. It is shown that if an ideal class \u2102 maps [r,s,t] to [u,v,w] then the class \u2102-1 maps [-r,-s,-t] to [-u,-v,-w]. From these two facts, several results follow. If \u2102 maps [r,s,o] to [u,v,w] then \u2102 has order 2 if one of u, v or w is 0. If \u2102 maps [r,s,o] to [u,v,v] then \u2102 has order 4 and the class \u21022 contains \u21192. If \u2102 maps [r,s,o] to [u,v,w] then \u2102-1 maps [r,s,o] to [-u,-v,-w]. If m can be written as a sum of two squares then a class \u2102 is the square of another class (i.e. \u2102 is in the principal genus) if \u2102 maps some bundle [u,v,w] to [-u,-v,-w].
\r\n\r\nWe apply these results to the following problem; given an odd prime p and an odd integer n, in which ring class groups are the prime divisors of p regular ideals in classes of order n? It is shown that the number of such ring class groups having discriminant -4m where m is a sum of two squares is related to the class number h(-4p) of the field \u211a(\u221a-p). For n = 3 the number is given by
\r\n\r\n1/16 f(p)h(-4p) - 6h(-4p) + 2 if p \u2261 1 mod 4
\r\n1/8 f(p)h(-4p) - 6h(-4p) if p \u2261 3 mod 8
\r\n0 if p \u2261 7 mod 8
\r\n\r\nHere f(p) is the number of ways to write p as a sum of 4 squares plus the number of ways to write 4p as a sum of 4 odd squares. A simple algorithm for producing the discriminants of all such ring class groups is given. Similar, but more complicated formulas hold for odd numbers n greater than 3.
", "doi": "10.7907/2G8A-Y604", "publication_date": "1981", "thesis_type": "phd", "thesis_year": "1981" }, { "id": "thesis:10133", "collection": "thesis", "collection_id": "10133", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:04072017-144744158", "primary_object_url": { "basename": "Parker_JA_1976.pdf", "content": "final", "filesize": 11186772, "license": "other", "mime_type": "application/pdf", "url": "/10133/1/Parker_JA_1976.pdf", "version": "v2.0.0" }, "type": "thesis", "title": "The Matrix Equation F(A)X - XA = O", "author": [ { "family_name": "Parker", "given_name": "Joseph A. Jr.", "clpid": "Parker-Joseph-A" } ], "thesis_advisor": [ { "family_name": "Taussky-Todd", "given_name": "Olga", "clpid": "Taussky-Todd-O" } ], "thesis_committee": [ { "family_name": "Unknown", "given_name": "Unknown" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "In this work, all matrices are assumed to have complex entries. The cases of\r\nF(A) - XA = O where F(A) is a polynomial over C in A and\r\nF(A) = (A*)-1 are investigated. Canonical forms are derived for\r\nsolutions X to these equations. Other results are given for matrices\r\nof the form A-1A*.
\r\n\r\nLet a set solutions {Xi} be called a tower if Xi+1 = F(Xi).\r\nIt is shown that towers occur for all nonsingular solutions of\r\n(A*)-1X - XA = O if and only if A is normal. In contrast to this, there\r\nis no polynomial for which only normal matrices A imply the existence of\r\ntowers for all solutions X of P(A)X - XA = O. On the other hand, conditions\r\nare given for polynomials P, dependent upon spectrum of A, for\r\nwhich only diagonalizable matrices A imply the existence of towers for all solutions\r\nX of P(A)X - XA = O.
\r\n", "doi": "10.7907/X6W2-T505", "publication_date": "1976", "thesis_type": "phd", "thesis_year": "1976" }, { "id": "thesis:9710", "collection": "thesis", "collection_id": "9710", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:05092016-130648083", "primary_object_url": { "basename": "Loewy_r_1972.pdf", "content": "final", "filesize": 15739845, "license": "other", "mime_type": "application/pdf", "url": "/9710/1/Loewy_r_1972.pdf", "version": "v2.0.0" }, "type": "thesis", "title": "On the Lyapunov Transformation for Stable Matrices", "author": [ { "family_name": "Loewy", "given_name": "Raphael", "clpid": "Loewy-Raphael" } ], "thesis_advisor": [ { "family_name": "Taussky-Todd", "given_name": "Olga", "clpid": "Taussky-Todd-O" } ], "thesis_committee": [ { "family_name": "Unknown", "given_name": "Unknown" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "The matrices studied here are positive stable (or briefly stable). These are matrices, real or complex, whose eigenvalues have positive real parts. A theorem of Lyapunov states that A is stable if and only if there exists H \u02c3 0 such that AH + HA* = I. Let A be a stable matrix. Three aspects of the Lyapunov transformation LA :H \u2192 AH + HA* are discussed.
\r\n\r\n1. Let C1 (A) = {AH + HA* :H \u2265 0} and C2 (A) = {H: AH+HA* \u2265 0}. The problems of determining the cones C1(A) and C2(A) are still unsolved. Using solvability theory for linear equations over cones it is proved that C1(A) is the polar of C2(A*), and it is also shown that C1 (A) = C1(A-1). The inertia assumed by matrices in C1(A) is characterized.
\r\n\r\n2. The index of dissipation of A was defined to be the maximum number of equal eigenvalues of H, where H runs through all matrices in the interior of C2(A). Upper and lower bounds, as well as some properties of this index, are given.
\r\n\r\n3. We consider the minimal eigenvalue of the Lyapunov transform AH+HA*, where H varies over the set of all positive semi-definite matrices whose largest eigenvalue is less than or equal to one. Denote it by \u03c8(A). It is proved that if A is Hermitian and has eigenvalues \u03bc1 \u2265 \u03bc2\u2026\u2265 \u03bcn \u02c3 0, then \u03c8(A) = -(\u03bc1-\u03bcn)2/(4(\u03bc1 + \u03bcn)). The value of \u03c8(A) is also determined in case A is a normal, stable matrix. Then \u03c8(A) can be expressed in terms of at most three of the eigenvalues of A. If A is an arbitrary stable matrix, then upper and lower bounds for \u03c8(A) are obtained.
\r\n", "doi": "10.7907/EXGT-0968", "publication_date": "1972", "thesis_type": "phd", "thesis_year": "1972" }, { "id": "thesis:17494", "collection": "thesis", "collection_id": "17494", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:06272025-174950440", "primary_object_url": { "basename": "Uhlig_FD_1972.pdf", "content": "final", "filesize": 31847665, "license": "other", "mime_type": "application/pdf", "url": "/17494/1/Uhlig_FD_1972.pdf", "version": "v2.0.0" }, "type": "thesis", "title": "A Study of the Canonical Form for a Pair of Real Symmetric Matrices and Applications to Pencils and to Pairs of Quadratic Forms", "author": [ { "family_name": "Uhlig", "given_name": "Frank Detlev", "orcid": "0000-0002-7495-5753", "clpid": "Uhlig-Frank-Detlev" } ], "thesis_advisor": [ { "family_name": "Taussky-Todd", "given_name": "Olga", "clpid": "Taussky-Todd-O" } ], "thesis_committee": [ { "family_name": "Unknown", "given_name": "Unknown" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "A pair of real symmetric matrices S and T is called a nonsingular pair\r\nif Sis nonsingular. A new treatment for obtaining the classical canonical\r\npair form for a nonsingular pair is obtained by the use of results on\r\ncommuting matrices and by elementary matrix algebra.\r\nThis canonical form is used to obtain formulas for an arbitrary real\r\nn X n matrix A that relate the dimensions of both the space N of real\r\nI symmetric matrices T such that AT = TA and the space of products AT\r\nsuch that AT is symmetric to the real Jordan normal form of A. The\r\nfirst formula expresses a previously found result in a simpler way\r\nwhile the second one is new. These formulas are then applied to prove\r\nanew the known result that A is nonderogatory iff dim N = n.\r\nSimultaneous diagonalization of two real symmetric matrices has been\r\nof interest. For instance it has been shown that if the quadratic forms\r\nassociated with Sand T (of dimensions greater than 2) do not vanish\r\nsimultaneously, then S and T can be diagonalized simultaneously by a\r\nreal congruence transformation. This subject is generalized here to\r\nthe study of the following two problems:
\r\n\r\n1) The finest simultaneous block diagonal structure for nonsingular\r\npairs,
\r\n\r\n2) common annihilating vectors of the corresponding quadratic forms.\r\nThe proofs are obtained here by algebraic means. Results:\r\nad 1) A simultaneous block diagonalization X' TX= diag(A1,...,Ak\r\nand X'TX = diag(B1,...,Bk) with dim Ai = dim Bi and X nonsingular\r\nis the finest simultaneous block diagonalization of a nonsingular pair\r\nS and T, if k is maximal. In this finest diagonalization the sizes of\r\nthe blocks Ai are uniquely determined (up to permutations) by any set\r\nof generators of the pencil P(S,T) = {aS + bT]a,b \u03f5 R}. The number k\r\nand the sizes of the diagonal blocks are also derived from the factorization\r\nover C of f(\u03bb,\u00b5) = det(\u03bbS + \u00b5T) for \u03bb, \u00b5 \u03f5 R.\r\nad 2) Knowing the real Jordan normal form of S-1T for a nonsingular\r\npair S and T we compute the maximal number m of linearly independent\r\nvectors that are simultaneously annihilated by the corresponding quadratic\r\nforms. Conversely, knowing m for two quadratic forms we deduce\r\nthe first simultaneous block diagonal structure of S and T, the corresponding\r\npair of real symmetric matrices. This is used to give new\r\nsufficient conditions for S and T to be simultaneously diagonalizable.
", "doi": "10.7907/pfpv-ty08", "publication_date": "1972", "thesis_type": "phd", "thesis_year": "1972" }, { "id": "thesis:9680", "collection": "thesis", "collection_id": "9680", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:04182016-160159252", "primary_object_url": { "basename": "Johnson_cr_1972.pdf", "content": "final", "filesize": 18736301, "license": "other", "mime_type": "application/pdf", "url": "/9680/1/Johnson_cr_1972.pdf", "version": "v2.0.0" }, "type": "thesis", "title": "Matrices whose Hermitian Part is Positive Definite", "author": [ { "family_name": "Johnson", "given_name": "Charles Royal", "clpid": "Johnson-Charles-Royal" } ], "thesis_advisor": [ { "family_name": "Taussky-Todd", "given_name": "Olga", "clpid": "Taussky-Todd-O" } ], "thesis_committee": [ { "family_name": "Unknown", "given_name": "Unknown" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "We are concerned with the class \u220fn of nxn complex matrices A for which the Hermitian part H(A) = A+A*/2 is positive definite.
\r\n\r\nVarious connections are established with other classes such as the stable, D-stable and dominant diagonal matrices. For instance it is proved that if there exist positive diagonal matrices D, E such that DAE is either row dominant or column dominant and has positive diagonal entries, then there is a positive diagonal F such that FA \u03f5 \u220fn.
\r\n\r\nPowers are investigated and it is found that the only matrices A for which Am \u03f5 \u220fn for all integers m are the Hermitian elements of \u220fn. Products and sums are considered and criteria are developed for AB to be in \u220fn.
\r\n\r\nSince \u220fn n is closed under inversion, relations between H(A)-1 and H(A-1) are studied and a dichotomy observed between the real and complex cases. In the real case more can be said and the initial result is that for A \u03f5 \u220fn, the difference H(adjA) - adjH(A) \u2265 0 always and is \u02c3 0 if and only if S(A) = A-A*/2 has more than one pair of conjugate non-zero characteristic roots. This is refined to characterize real c for\r\nwhich cH(A-1) - H(A)-1 is positive definite.
\r\n\r\nThe cramped (characteristic roots on an arc of less than 180\u00b0) unitary matrices are linked to \u220fn and characterized in several ways via products of the form A -1A*.
\r\n\r\nClassical inequalities for Hermitian positive definite matrices are studied in \u220fn and for Hadamard's inequality two types of generalizations are given. In the first a large subclass of \u220fn in which the precise statement of Hadamardis inequality holds is isolated while in another large subclass its reverse is shown to hold. In the second Hadamard's inequality is weakened in such a way that it holds throughout \u220fn. Both approaches contain the original Hadamard inequality as a special case.
\r\n\r\n", "doi": "10.7907/ZXNF-SB10", "publication_date": "1972", "thesis_type": "phd", "thesis_year": "1972" }, { "id": "thesis:10114", "collection": "thesis", "collection_id": "10114", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:03282017-155524180", "primary_object_url": { "basename": "Maurer_DE_1969.pdf", "content": "final", "filesize": 11133682, "license": "other", "mime_type": "application/pdf", "url": "/10114/1/Maurer_DE_1969.pdf", "version": "v2.0.0" }, "type": "thesis", "title": "Modules with Integral Discriminant Matrix", "author": [ { "family_name": "Maurer", "given_name": "Donald Eugene", "clpid": "Maurer-Donald-Eugene" } ], "thesis_advisor": [ { "family_name": "Taussky-Todd", "given_name": "Olga", "clpid": "Taussky-Todd-O" } ], "thesis_committee": [ { "family_name": "Unknown", "given_name": "Unknown" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "Let F be a field which admits a Dedekind set of spots (see\r\nO'Meara, Introduction to Quadratic Forms) and such that the integers\r\nZF of F form a principal ideal domain. Let K|F be a separable\r\nalgebraic extension of F of degree n. If M is a ZF-module contained\r\nin K, and \u03c31, \u03c32, ..., \u03c3n is a ZF-basis for M, the matrix D(\u03c3) = (traceK|F(\u03c3i\u03c3j)) is called a discriminant matrix. We study modules which have an integral discriminant matrix. When F is the rational field, we are able to obtain necessary and sufficient conditions on det D(\u03c3) in order that M be properly contained in a larger module having an integral discriminant matrix. This is equivalent to determining when the corresponding quadratic form
\r\n\r\nf = \u03a3ij aijxixj (aij = aaji),\r\n\r\n\r\nwith integral matrix (aij) can be obtained from another such form, with\r\nlarger determinant, by an integral transformation.
\r\n\r\nThese two main results are then applied to characterize normal\r\nalgebraic extensions K of the rationals in which ZK is maximal with\r\nrespect to having an integral discriminant matrix.
", "doi": "10.7907/BQG6-4P65", "publication_date": "1969", "thesis_type": "phd", "thesis_year": "1969" }, { "id": "thesis:9554", "collection": "thesis", "collection_id": "9554", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:02012016-080841282", "primary_object_url": { "basename": "Davis_dl_1969.pdf", "content": "final", "filesize": 19736080, "license": "other", "mime_type": "application/pdf", "url": "/9554/1/Davis_dl_1969.pdf", "version": "v2.0.0" }, "type": "thesis", "title": "On the Distribution of the Signs of the Conjugates of the Cyclotomic Units in the Maximal Real Subfield of the qth Cyclotomic Field, q a Prime", "author": [ { "family_name": "Davis", "given_name": "Daniel Lee", "clpid": "Davis-Daniel-Lee" } ], "thesis_advisor": [ { "family_name": "Taussky-Todd", "given_name": "Olga", "clpid": "Taussky-Todd-O" } ], "thesis_committee": [ { "family_name": "Unknown", "given_name": "Unknown" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "Let F = \u01ea(\u03b6 + \u03b6 \u20131) be the maximal real subfield of the cyclotomic field \u01ea(\u03b6) where \u03b6 is a primitive qth root of unity and q is an odd rational prime. The numbers u1=-1, uk=(\u03b6k-\u03b6-k)/(\u03b6-\u03b6-1), k=2,\u2026,p, p=(q-1)/2, are units in F and are called the cyclotomic units. In this thesis the sign distribution of the conjugates in F of the cyclotomic units is studied.
\r\n\r\nLet G(F/\u01ea) denote the Galoi's group of F over \u01ea, and let V denote the units in F. For each \u03c3\u03f5 G(F/\u01ea) and \u03bc\u03f5V define a mapping sgn\u03c3: V\u2192GF(2) by sgn\u03c3(\u03bc) = 1 iff \u03c3(\u03bc) \u02c2 0 and sgn\u03c3(\u03bc) = 0 iff \u03c3(\u03bc) \u02c3 0. Let {\u03c31, ... , \u03c3p} be a fixed ordering of G(F/\u01ea). The matrix Mq=(sgn\u03c3j(vi) ) , i, j = 1, ... , p is called the matrix of cyclotomic signatures. The rank of this matrix determines the sign distribution of the conjugates of the cyclotomic units. The matrix of cyclotomic signatures is associated with an ideal in the ring GF(2) [x] / (xp+ 1) in such a way that the rank of the matrix equals the GF(2)-dimension of the ideal. It is shown that if p = (q-1)/ 2 is a prime and if 2 is a primitive root mod p, then Mq is non-singular. Also let p be arbitrary, let \u2113 be a primitive root mod q and let L = {i | 0 \u2264 i \u2264 p-1, the least positive residue of defined by \u2113i mod q is greater than p}. Let Hq(x) \u03f5 GF(2)[x] be defined by Hq(x) = g. c. d. ((\u03a3 xi/I \u03f5 L) (x+1) + 1, xp + 1). It is shown that the rank of Mq equals the difference p - degree Hq(x).
\r\n\r\nFurther results are obtained by using the reciprocity theorem of class field theory. The reciprocity maps for a certain abelian extension of F and for the infinite primes in F are associated with the signs of conjugates. The product formula for the reciprocity maps is used to associate the signs of conjugates with the reciprocity maps at the primes which lie above (2). The case when (2) is a prime in F is studied in detail. Let T denote the group of totally positive units in F. Let U be the group generated by the cyclotomic units. Assume that (2) is a prime in F and that p is odd. Let F(2) denote the completion of F at (2) and let V(2) denote the units in F(2). The following statements are shown to be equivalent. 1) The matrix of cyclotomic signatures is non-singular. 2) U\u2229T = U2. 3) U\u2229F2(2) = U2. 4) V(2)/ V(2)2 = \u02c2v1 V(2)2\u02c3 \u0298\u2026\u0298\u02c2vp V(2)2\u02c3 \u0298 \u02c23V(2)2\u02c3.
\r\n\r\nThe rank of Mq was computed for 5\u2264q\u2264929 and the results appear in tables. On the basis of these results and additional calculations the following conjecture is made: If q and p = (q -1)/ 2 are both primes, then Mq is non-singular.
\r\n", "doi": "10.7907/YG5G-HE68", "publication_date": "1969", "thesis_type": "phd", "thesis_year": "1969" }, { "id": "thesis:9165", "collection": "thesis", "collection_id": "9165", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:09222015-114033548", "primary_object_url": { "basename": "Gaines_fj_1966.pdf", "content": "final", "filesize": 14687320, "license": "other", "mime_type": "application/pdf", "url": "/9165/1/Gaines_fj_1966.pdf", "version": "v2.0.0" }, "type": "thesis", "title": "Some Generalizations of Commutativity for Linear Transformations on a Finite Dimensional Vector Space", "author": [ { "family_name": "Gaines", "given_name": "Fergus John", "clpid": "Gaines-Fergus-John" } ], "thesis_advisor": [ { "family_name": "Taussky-Todd", "given_name": "Olga", "clpid": "Taussky-Todd-O" } ], "thesis_committee": [ { "family_name": "Unknown", "given_name": "Unknown" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "Let L be the algebra of all linear transformations on an n-dimensional vector space V over a field F and let A, B, \u0190L. Let Ai+1 = AiB - BAi, i = 0, 1, 2,\u2026, with A = Ao. Let fk (A, B; \u03c3) = A2K+1 - \u03c31A2K-1 + \u03c32A2K-3 -\u2026 +(-1)K\u03c3KA1 where \u03c3 = (\u03c31, \u03c32,\u2026, \u03c3K), \u03c3i belong to F and K = k(k-1)/2. Taussky and Wielandt [Proc. Amer. Math. Soc., 13(1962), 732-735] showed that fn(A, B; \u03c3) = 0 if \u03c3i is the ith elementary symmetric function of (\u03b24- \u03b2s)2, 1 \u2264 r \u02c2 s \u2264 n, i = 1, 2, \u2026, N, with N = n(n-1)/2, where \u03b24 are the characteristic roots of B. In this thesis we discuss relations involving fk(X, Y; \u03c3) where X, Y \u0190 L and 1 \u2264 k \u02c2 n. We show: 1. If F is infinite and if for each X \u0190 L there exists \u03c3 so that fk(A, X; \u03c3) = 0 where 1 \u2264 k \u02c2 n, then A is a scalar transformation. 2. If F is algebraically closed, a necessary and sufficient condition that there exists a basis of V with respect to which the matrices of A and B are both in block upper triangular form, where the blocks on the diagonals are either one- or two-dimensional, is that certain products X1, X2\u2026Xr belong to the radical of the algebra generated by A and B over F, where Xi has the form f2(A, P(A,B); \u03c3), for all polynomials P(x, y). We partially generalize this to the case where the blocks have dimensions \u2264 k. 3. If A and B generate L, if the characteristic of F does not divide n and if there exists \u03c3 so that fk(A, B; \u03c3) = 0, for some k with 1 \u2264 k \u02c2 n, then the characteristic roots of B belong to the splitting field of gk(w; \u03c3) = w2K+1 - \u03c31w2K-1 + \u03c32w2K-3 - \u2026. +(-1)K \u03c3Kw over F. We use this result to prove a theorem involving a generalized form of property L [cf. Motzkin and Taussky, Trans. Amer. Math. Soc., 73(1952), 108-114]. 4. Also we give mild generalizations of results of McCoy [Amer. Math. Soc. Bull., 42(1936), 592-600] and Drazin [Proc. London Math. Soc., 1(1951), 222-231].
", "doi": "10.7907/PBNM-BZ75", "publication_date": "1966", "thesis_type": "phd", "thesis_year": "1966" }, { "id": "thesis:9157", "collection": "thesis", "collection_id": "9157", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:09172015-140824031", "primary_object_url": { "basename": "Bender_ea_1966.pdf", "content": "final", "filesize": 19698362, "license": "other", "mime_type": "application/pdf", "url": "/9157/1/Bender_ea_1966.pdf", "version": "v2.0.0" }, "type": "thesis", "title": "Symmetric Representations of an Integral Domain over a Subdomain", "author": [ { "family_name": "Bender", "given_name": "Edward Anton", "clpid": "Bender-Edward-Anton" } ], "thesis_advisor": [ { "family_name": "Taussky-Todd", "given_name": "Olga", "clpid": "Taussky-Todd-O" } ], "thesis_committee": [ { "family_name": "Unknown", "given_name": "Unknown" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "Let F(\u03b8) be a separable extension of degree n of a field F. Let \u0394 and D be integral domains with quotient fields F(\u03b8) and F respectively. Assume that \u0394 \u1d1d D. A mapping \u03c6 of \u0394 into the n x n D matrices is called a \u0394/D rep if (i) it is a ring isomorphism and (ii) it maps d onto dIn whenever d \u03f5 D. If the matrices are also symmetric, \u03c6 is a \u0394/D symrep.
\r\n\r\nEvery \u0394/D rep can be extended uniquely to an F(\u03b8)/F rep. This extension is completely determined by the image of \u03b8. Two \u0394/D reps are called equivalent if the images of \u03b8 differ by a D unimodular similarity. There is a one-to-one correspondence between classes of \u0394/D reps and classes of \u0394 ideals having an n element basis over D.
\r\n\r\nThe condition that a given \u0394/D rep class contain a \u0394/D symrep can be phrased in various ways. Using these formulations it is possible to (i) bound the number of symreps in a given class, (ii) count the number of symreps if F is finite, (iii) establish the existence of an F(\u03b8)/F symrep when n is odd, F is an algebraic number field, and F(\u03b8) is totally real if F is formally real (for n = 3 see Sapiro, \u201cCharacteristic polynomials of symmetric matrices\u201d Sibirsk. Mat. \u017d. 3 (1962) pp. 280-291), and (iv) study the case D = Z, the integers (see Taussky, \u201cOn matrix classes corresponding to an ideal and its inverse\u201d Illinois J. Math. 1 (1957) pp. 108-113 and Faddeev, \u201cOn the characteristic equations of rational symmetric matrices\u201d Dokl. Akad. Nauk SSSR 58 (1947) pp. 753-754).
\r\n\r\nThe case D = Z and n = 2 is studied in detail. Let \u0394\u2019 be an integral domain also having quotient field F(\u03b8) and such that \u0394\u2019 \u1d1d \u0394. Let \u03c6 be a \u0394/Z symrep. A method is given for finding a \u0394\u2019/Z symrep \u0298 such that the \u0394\u2019 ideal class corresponding to the class of \u0298 is an extension to \u0394\u2019 of the \u0394 ideal class corresponding to the class of \u03c6. The problem of finding all \u0394/Z symreps equivalent to a given one is studied.
\r\n", "doi": "10.7907/KDAH-VV38", "publication_date": "1966", "thesis_type": "phd", "thesis_year": "1966" }, { "id": "thesis:3570", "collection": "thesis", "collection_id": "3570", "cite_using_url": "https://resolver.caltech.edu/CaltechETD:etd-09172002-111103", "primary_object_url": { "basename": "Foster_l_1964.pdf", "content": "final", "filesize": 2944619, "license": "other", "mime_type": "application/pdf", "url": "/3570/1/Foster_l_1964.pdf", "version": "v2.0.0" }, "type": "thesis", "title": "On the Characteristic Roots of the Product of Certain Rational Integral Matrices of Order Two", "author": [ { "family_name": "Foster", "given_name": "Lorraine Lois Turnbull", "clpid": "Foster-Lorraine-Lois-Turnbull" } ], "thesis_advisor": [ { "family_name": "Taussky-Todd", "given_name": "Olga", "clpid": "Taussky-Todd-O" } ], "thesis_committee": [ { "family_name": "Unknown", "given_name": "Unknown" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document.\r\n\r\nLet N(p,q) denote the companion matrix of x[superscript 2] + px + q, for rational integers p and q, and let M(p,q)=N(p,q)(N(p,q))'. Further let F(M(p,q)) and F(N(p,q)) denote the fields generated by the characteristic roots of M(p,q) and N(p,q) over the rational field, R. This thesis is concerned with F(M(p,q)), especially in relation to F(N(p,q)). The principal results obtained are outlined as follows:\r\n\r\nLet S be the set of square-free integers which are sums of two squares. Then F(M(p,q)) is of the form R[...], where c [...] S. Further, F(M(p,q)) = R if and only if pq = 0. Suppose c [...] S. Then there exist infinitely many distinct pairs of integers (p,q) such that F(M(p,q)) = R[...].\r\n\r\nFurther, if c [...] S., there exists a sequence {(p[subscript n],q[subscript n])} of distinct pairs of integers such that F(N(p[subscript n],q[subscript n])) =R[...], and F(MN(p[subscript n],q[subscript n])) = R[...], where the d[subscript n] are some integers such that c,d[subscript n] = 1. If c [...] S and c is odd or c = 2, there exists a sequence {(p'[subscript n],q'[subscript n])} of distinct pairs of integers such that F(M(p'[subscript n],q'[subscript n)) = R[...] and F(N(p'[subscript n],q'[subscript n)) = R[...], for some integers d'[subscript n] such that (c,d'[subscript n]) = 1.\r\n\r\nThere are five known pairs of integers (p,q), with pq [not equalling] 0 and q [not equalling 1, such that F(M(p,q)) and F(N(p,q)) coincide. For q [...] and for certain odd integers q, the fields F(M(p,q)) and F(N(p,q)) cannot coincide for any integers p.\r\n\r\nFinally, for any integer p [not equalling] (or q [not equalling] 0, -1) there exist at most a finite number of integers q (or p) such that the two fields coincide.", "doi": "10.7907/JE7E-1393", "publication_date": "1964", "thesis_type": "phd", "thesis_year": "1964" }, { "id": "thesis:2608", "collection": "thesis", "collection_id": "2608", "cite_using_url": "https://resolver.caltech.edu/CaltechETD:etd-06152006-083527", "primary_object_url": { "basename": "Hobby_c_1960.pdf", "content": "final", "filesize": 2865408, "license": "other", "mime_type": "application/pdf", "url": "/2608/1/Hobby_c_1960.pdf", "version": "v2.0.0" }, "type": "thesis", "title": "The Derived Series of a p-Group", "author": [ { "family_name": "Hobby", "given_name": "Charles Ray", "clpid": "Hobby-Charles-Ray" } ], "thesis_advisor": [ { "family_name": "Taussky-Todd", "given_name": "Olga", "clpid": "Taussky-Todd-O" }, { "family_name": "Zassenhaus", "given_name": "Hans", "clpid": "Zassenhaus-Hans" } ], "thesis_committee": [ { "family_name": "Unknown", "given_name": "Unknown" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document.\r\n\r\nOlga Taussky (see W. Magnus, Math. Ann. vol. 111 (1935)) posed the problem of determining whether there is an infinite chain of p-groups G1, G2,..., such that G1 is abelian, [...], and [...] where [...] is the nth derived group of [...]. N. Ito (Nagoya Math. J., vol. 1, (1950)) constructed such a chain for p > 2 and G1 of type (p,p,p). It is shown (by an explicit construction) that if p > 2 there is a chain of the required kind for G1 any non-cyclic abelian p-group. If p = 2 there is a chain of the required kind if G1 contains a subgroup of type [...], of type [...], of type [...], or of type (2,2,2,2,2). As a consequence, for p > 2 it is impossible to estimate the length of the derived series of a non-abelian p-group G from the type of [...]. This gives a negative answer (for p > 2) to a question posed by O. Taussky (Research Problem 9, Bull. Amer. Math. Soc. vol. 64 (1958) pp. 124).", "doi": "10.7907/QY7G-Q706", "publication_date": "1960", "thesis_type": "phd", "thesis_year": "1960" }, { "id": "thesis:2820", "collection": "thesis", "collection_id": "2820", "cite_using_url": "https://resolver.caltech.edu/CaltechETD:etd-07072006-084911", "primary_object_url": { "basename": "Thompson_rc_1960.pdf", "content": "final", "filesize": 3431428, "license": "other", "mime_type": "application/pdf", "url": "/2820/1/Thompson_rc_1960.pdf", "version": "v2.0.0" }, "type": "thesis", "title": "Commutators in the Special and General Linear Groups", "author": [ { "family_name": "Thompson", "given_name": "Robert Charles", "clpid": "Thompson-Robert-Charles" } ], "thesis_advisor": [ { "family_name": "Taussky-Todd", "given_name": "Olga", "clpid": "Taussky-Todd-O" } ], "thesis_committee": [ { "family_name": "Unknown", "given_name": "Unknown" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document.\r\n\r\nLet GL(n, K) denote the multiplicative group of all non-singular nxn matrices with coefficients in a field K; SL(n, K) the subgroup of GL(n, K) consisting of all matrices with determinant unity; C(n, K) the centre of SL(n, K); PSL(n, K) the factor group SL(n, K)/C(n, K); I(n) the nxn identity matrix; GF(pn) the finite field with pn elements. We determine when every element of SL(n, K) is a commutator of SL(n, K) or of GL(n, K). Theorem 1. Let A [...] SL(n, K). Then it follows that A is a commutator [...] of SL(n, K) unless: (i) n = 2 and K = GF(2); (ii) n = 2 and K = GF(3); or (iii) K has characteristic zero and A = [...] where a is a primitive nth root of unity in K and n [...] 2 (mod 4). In case (i), SL(2, GF(2)) properly contains its commutator subgroup. In case (ii), SL(2, GF(3)) properly contains its commutator subgroup. Furthermore, every element of SL(2, GF(3)) is a commutator of GL(2, GF(3)). In case (iii), [...] is always a commutator of GL(n, K). Moreover, aIn is a commutator of SL(n, K) when, and only when, the equation -1 = x2 + y2 has a solution x, y [...] K. Hence: Theorem 2. Whenever PSL(n, K) is simple, every element of PSL(n, K) is a commutator of PSL(n, K). Theorem 1 simplifies and extends results due to K. Shoda (Jap. J. Math., 13 (1936), p. 361-365; J. Math. Soc. of Japan, 3 (1951), p. 78-81). Theorem 2 supports the suggestion made by O. Ore (Proc. Amer. Math. Soc., 2 (1951), p. 307-314) that in a finite simple group, every element is a commutator.", "doi": "10.7907/3BQM-3542", "publication_date": "1960", "thesis_type": "phd", "thesis_year": "1960" } ]