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A Caltech Library Repository Feedhttp://www.rssboard.org/rss-specificationpython-feedgenenFri, 08 Dec 2023 22:31:35 +0000The Derived Series of a p-Group
https://resolver.caltech.edu/CaltechETD:etd-06152006-083527
Authors: Hobby, Charles Ray
Year: 1960
DOI: 10.7907/QY7G-Q706
NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document.
Olga 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).https://thesis.library.caltech.edu/id/eprint/2608Commutators in the Special and General Linear Groups
https://resolver.caltech.edu/CaltechETD:etd-07072006-084911
Authors: Thompson, Robert Charles
Year: 1960
DOI: 10.7907/3BQM-3542
NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document.
Let 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.https://thesis.library.caltech.edu/id/eprint/2820Some generalizations of commutativity for linear transformations on a finite dimensional vector space
https://resolver.caltech.edu/CaltechTHESIS:09222015-114033548
Authors: Gaines, Fergus John
Year: 1966
DOI: 10.7907/PBNM-BZ75
<p>Let <i>L</i> be the algebra of all linear transformations on an n-dimensional vector space V over a field <i>F</i> and let A, B, Ɛ<i>L</i>. Let A<sub>i+1</sub> = A<sub>i</sub>B - BA<sub>i</sub>, i = 0, 1, 2,…, with A = A<sub>o</sub>. Let f<sub>k</sub> (A, B; σ) = A<sub>2K+1</sub> - <sup>σ</sup>1<sup>A</sup>2K-1 <sup>+</sup> <sup>σ</sup>2<sup>A</sup>2K-3 -… +(-1)<sup>K</sup>σ<sub>K</sub>A<sub>1</sub> where σ = (σ<sub>1</sub>, σ<sub>2</sub>,…, σ<sub>K</sub>), σ<sub>i</sub> belong to <i>F</i> and K = k(k-1)/2. Taussky and Wielandt [Proc. Amer. Math. Soc., 13(1962), 732-735] showed that f<sub>n</sub>(A, B; σ) = 0 if σ<sub>i</sub> is the i<sup>th</sup> elementary symmetric function of (β<sub>4</sub>- β<sub>s</sub>)<sup>2</sup>, 1 ≤ r ˂ s ≤ n, i = 1, 2, …, N, with N = n(n-1)/2, where β<sub>4</sub> are the characteristic roots of B. In this thesis we discuss relations involving f<sub>k</sub>(X, Y; σ) where X, Y Ɛ <i>L</i> and 1 ≤ k ˂ n. We show: 1. If <i>F</i> is infinite and if for each X Ɛ <i>L</i> there exists σ so that f<sub>k</sub>(A, X; σ) = 0 where 1 ≤ k ˂ n, then A is a scalar transformation. 2. If <i>F</i> 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 X<sub>1</sub>, X<sub>2</sub>…X<sub>r</sub> belong to the radical of the algebra generated by A and B over <i>F</i>, where X<sub>i</sub> has the form f<sub>2</sub>(A, P(A,B); σ), for all polynomials P(x, y). We partially generalize this to the case where the blocks have dimensions ≤ k. 3. If A and B generate <i>L</i>, if the characteristic of <i>F</i> does not divide n and if there exists σ so that f<sub>k</sub>(A, B; σ) = 0, for some k with 1 ≤ k ˂ n, then the characteristic roots of B belong to the splitting field of g<sub>k</sub>(w; σ) = w<sup>2K+1</sup> - σ<sub>1</sub>w<sup>2K-1</sup> + σ<sub>2</sub>w<sup>2K-3</sup> - …. +(-1)<sup>K</sup> σ<sub>K</sub>w over <i>F</i>. 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]. </p>https://thesis.library.caltech.edu/id/eprint/9165Symmetric Representations of an Integral Domain over a Subdomain
https://resolver.caltech.edu/CaltechTHESIS:09172015-140824031
Authors: Bender, Edward Anton
Year: 1966
DOI: 10.7907/KDAH-VV38
<p>Let F(θ) be a separable extension of degree n of a field F. Let Δ and D be integral domains with quotient fields F(θ) and F respectively. Assume that Δ <u>ᴝ</u> D. A mapping φ of Δ into the n x n D matrices is called a Δ/D rep if (i) it is a ring isomorphism and (ii) it maps d onto dI<sub>n</sub> whenever d ϵ D. If the matrices are also symmetric, φ is a Δ/D symrep.</p>
<p>Every Δ/D rep can be extended uniquely to an F(θ)/F rep. This extension is completely determined by the image of θ. Two Δ/D reps are called equivalent if the images of θ differ by a D unimodular similarity. There is a one-to-one correspondence between classes of Δ/D reps and classes of Δ ideals having an n element basis over D. </p>
<p>The condition that a given Δ/D rep class contain a Δ/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(θ)/F symrep when n is odd, F is an algebraic number field, and F(θ) is totally real if F is formally real (for n = 3 see Sapiro, “Characteristic polynomials of symmetric matrices” Sibirsk. Mat. Ž. <u>3</u> (1962) pp. 280-291), and (iv) study the case D = Z, the integers (see Taussky, “On matrix classes corresponding to an ideal and its inverse” Illinois J. Math. <u>1</u> (1957) pp. 108-113 and Faddeev, “On the characteristic equations of rational symmetric matrices” Dokl. Akad. Nauk SSSR <u>58</u> (1947) pp. 753-754).</p>
<p>The case D = Z and n = 2 is studied in detail. Let Δ’ be an integral domain also having quotient field F(θ) and such that Δ’ <u>ᴝ</u> Δ. Let φ be a Δ/Z symrep. A method is given for finding a Δ’/Z symrep ʘ such that the Δ’ ideal class corresponding to the class of ʘ is an extension to Δ’ of the Δ ideal class corresponding to the class of φ. The problem of finding all Δ/Z symreps equivalent to a given one is studied. </p>
https://thesis.library.caltech.edu/id/eprint/9157Modules with Integral Discriminant Matrix
https://resolver.caltech.edu/CaltechTHESIS:03282017-155524180
Authors: Maurer, Donald Eugene
Year: 1969
DOI: 10.7907/BQG6-4P65
<p>Let F be a field which admits a Dedekind set of spots (see
O'Meara, Introduction to Quadratic Forms) and such that the integers
Z<sub>F</sub> of F form a principal ideal domain. Let K|F be a separable
algebraic extension of F of degree n. If M is a Z<sub>F</sub>-module contained
in K, and σ<sub>1</sub>, σ<sub>2</sub>, ..., σ<sub>n</sub> is a Z<sub>F</sub>-basis for M, the matrix D(σ) = (trace<sub>K|F</sub>(σ<sub>i</sub>σ<sub>j</sub>)) 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(σ) 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</p>
f = Σ<sub>ij</sub> a<sub>ij</sub>x<sub>i</sub>x<sub>j</sub> (a<sub>ij</sub> = aa<sub>ji</sub>),
<p>with integral matrix (a<sub>ij</sub>) can be obtained from another such form, with
larger determinant, by an integral transformation.</p>
<p>These two main results are then applied to characterize normal
algebraic extensions K of the rationals in which Z<sub>K</sub> is maximal with
respect to having an integral discriminant matrix.</p>https://thesis.library.caltech.edu/id/eprint/10114On 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
https://resolver.caltech.edu/CaltechTHESIS:02012016-080841282
Authors: Davis, Daniel Lee
Year: 1969
DOI: 10.7907/YG5G-HE68
<p>Let F = Ǫ(ζ + ζ<sup> –1</sup>) be the maximal real subfield of the cyclotomic field Ǫ(ζ) where ζ is a primitive qth root of unity and q is an odd rational prime. The numbers u<sub>1</sub>=-1, u<sub>k</sub>=(ζ<sup>k</sup>-ζ<sup>-k</sup>)/(ζ-ζ<sup>-1</sup>), k=2,…,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.</p>
<p>Let G(F/Ǫ) denote the Galoi's group of F over Ǫ, and let V denote the units in F. For each σϵ G(F/Ǫ) and μϵV define a mapping sgn<sub>σ</sub>: V→GF(2) by sgn<sub>σ</sub>(μ) = 1 iff σ(μ) ˂ 0 and sgn<sub>σ</sub>(μ) = 0 iff σ(μ) ˃ 0. Let {σ<sub>1</sub>, ... , σ<sub>p</sub>} be a fixed ordering of G(F/Ǫ). The matrix M<sub>q</sub>=(sgn<sub>σj</sub>(v<sub>i</sub>) ) , 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] / (x<sup>p</sup>+ 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 M<sub>q</sub> is non-singular. Also let p be arbitrary, let ℓ be a primitive root mod q and let L = {i | 0 ≤ i ≤ p-1, the least positive residue of defined by ℓ<sup>i</sup> mod q is greater than p}. Let H<sub>q</sub>(x) ϵ GF(2)[x] be defined by H<sub>q</sub>(x) = g. c. d. ((Σ x<sup>i</sup>/I ϵ L) (x+1) + 1, x<sup>p</sup> + 1). It is shown that the rank of M<sub>q</sub> equals the difference p - degree H<sub>q</sub>(x).</p>
<p>Further 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<sub>(2)</sub> denote the completion of F at (2) and let V<sub>(2)</sub> denote the units in F<sub>(2)</sub>. The following statements are shown to be equivalent. 1) The matrix of cyclotomic signatures is non-singular. 2) U∩T = U<sup>2</sup>. 3) U∩F<sup>2</sup><sub>(2)</sub> = U<sup>2</sup>. 4) V<sub>(2)</sub>/ V<sub>(2)</sub><sup>2</sup> = ˂<i>v</i><sub>1</sub> V<sub>(2)</sub><sup>2</sup>˃ ʘ…ʘ˂<i>v</i><sub>p</sub> V<sub>(2)</sub><sup>2</sup>˃ ʘ ˂3V<sub>(2)</sub><sup>2</sup>˃.</p>
<p>The rank of M<sub>q</sub> was computed for 5≤q≤929 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 M<sub>q</sub> is non-singular. </p>
https://thesis.library.caltech.edu/id/eprint/9554On the Lyapunov transformation for stable matrices
https://resolver.caltech.edu/CaltechTHESIS:05092016-130648083
Authors: Loewy, Raphael
Year: 1972
DOI: 10.7907/EXGT-0968
<p>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 ˃ 0 such that AH + HA* = I. Let A be a stable matrix. Three aspects of the Lyapunov transformation L<sub>A</sub> :H → AH + HA* are discussed.</p>
<p>1. Let C<sub>1</sub> (A) = {AH + HA* :H ≥ 0} and C<sub>2</sub> (A) = {H: AH+HA* ≥ 0}. The problems of determining the cones C<sub>1</sub>(A) and C<sub>2</sub>(A) are still unsolved. Using solvability theory for linear equations over cones it is proved that C<sub>1</sub>(A) is the polar of C<sub>2</sub>(A*), and it is also shown that C<sub>1</sub> (A) = C<sub>1</sub>(A<sup>-1</sup>). The inertia assumed by matrices in C<sub>1</sub>(A) is characterized. </p>
<p>2. 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 C<sub>2</sub>(A). Upper and lower bounds, as well as some properties of this index, are given.</p>
<p>3. 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 ψ(A). It is proved that if A is Hermitian and has eigenvalues μ<sub>1</sub> ≥ μ<sub>2</sub>…≥ μ<sub>n</sub> ˃ 0, then ψ(A) = -(μ<sub>1</sub>-μ<sub>n</sub>)<sup>2</sup>/(4(μ<sub>1</sub> + μ<sub>n</sub>)). The value of ψ(A) is also determined in case A is a normal, stable matrix. Then ψ(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 ψ(A) are obtained.</p>
https://thesis.library.caltech.edu/id/eprint/9710Matrices whose hermitian part is positive definite
https://resolver.caltech.edu/CaltechTHESIS:04182016-160159252
Authors: Johnson, Charles Royal
Year: 1972
DOI: 10.7907/ZXNF-SB10
<p>We are concerned with the class ∏<sub>n</sub> of nxn complex matrices A for which the Hermitian part H(A) = A+A*/2 is positive definite.</p>
<p>Various 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 ϵ ∏<sub>n</sub>. </p>
<p>Powers are investigated and it is found that the only matrices A for which A<sup>m</sup> ϵ ∏<sub>n</sub> for all integers m are the Hermitian elements of ∏<sub>n</sub>. Products and sums are considered and criteria are developed for AB to be in ∏<sub>n</sub>.</p>
<p>Since ∏<sub>n</sub> n is closed under inversion, relations between H(A)<sup>-1</sup> and H(A<sup>-1</sup>) 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 ϵ ∏<sub>n</sub>, the difference H(adjA) - adjH(A) ≥ 0 always and is ˃ 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
which cH(A<sup>-1</sup>) - H(A)<sup>-1</sup> is positive definite.</p>
<p>The cramped (characteristic roots on an arc of less than 180°) unitary matrices are linked to ∏<sub>n</sub> and characterized in several ways via products of the form A <sup>-1</sup>A*.</p>
<p>Classical inequalities for Hermitian positive definite matrices are studied in ∏<sub>n</sub> and for Hadamard's inequality two types of generalizations are given. In the first a large subclass of ∏<sub>n</sub> 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 ∏<sub>n</sub>. Both approaches contain the original Hadamard inequality as a special case. </p>
https://thesis.library.caltech.edu/id/eprint/9680Rational G-Circulants Satisfying the Matrix Equation A² = dI + λJ
https://resolver.caltech.edu/CaltechTHESIS:02022021-194446003
Authors: Lam, Clement Wing Hong
Year: 1974
DOI: 10.7907/7qh9-nn17
<p>A g-circulant is a square matrix of rational numbers in which each row is obtained from the preceding row by shifting the elements cyclically g columns to the right. This work studies g-circulants A which satisfy the matrix equation A<sup>2</sup> = dI + λJ, where I is the identity matrix and J is the matrix of 1's. Necessary and sufficient conditions are given for the existence of solutions when g = 1. The existence of (0,1) g-circulants satisfying A<sup>2</sup> = dI + λJ is shown to be equivalent to the existence of (v, k, λ, g)-addition sets, which are generalizations of difference sets. It is proved that there are no nontrivial (v, k, λ, 1)-addition sets. Some examples of (v, k, λ, g)-addition sets are given and the multiplier theorem for (v, k, λ, g)-addition sets is also proved.</p>https://thesis.library.caltech.edu/id/eprint/14071Numerical Ranges and Commutation Properties of Hilbert Space Operators
https://resolver.caltech.edu/CaltechTHESIS:02042021-163138095
Authors: Richard, Bruce Kent
Year: 1974
DOI: 10.7907/0j1k-nr94
<p>Application of the theory of numerical ranges to the study of commutation properties of operators is the purpose of the thesis.</p>
<p>For a complex, unital Banach algebra ℝ, T ∈ ℝ, the numerical range of T is V(ℝ, T) = {f(T) :f(1) = 1 = ∥f∥, f ∈ ℝ*}. This is a generalization and extension of the notion of the numerical range defined for a bounded operator T on the Hilbert space <b>H</b>: W(T) = {(Tx, x):x ∈ <b>H</b>, (x,x) = 1}. These numerical range concepts are used in studies of multiplicative commutators, derivations, and powers of accretive operators.</p>
<p>An extension of Frobenius' group commutator theorem is obtained: For T,A,B ∈ β(<b>H</b>), T = ABA<sup>-1</sup>B<sup>-1</sup>, AT = TA, A normal and 0 ∉W(B)<sup>-</sup> imply T = 1. Other extensions of the Frobenius theorem are proved and a special discussion is given about these results in the case <b>H</b> is finite dimensional. The sharpness of the results is also reviewed.</p>
<p>For X a Banach space, the numerical range of a derivation acting on β(X) is completely characterized. If Δ<sub>T</sub> is the derivation induced by T ∈ (β(X), then</p>
<p>V(β(β(X)), Δ<sub>T</sub>) = V(β(X),T) - V(β (X),T).</p>
<p>Normal elements of general Banach algebras are discussed. A consequence of an examination of derivations which are normal is a simple proof of the Fuglede- Putnam Theorem.</p>
<p>A theorem for matrices by C. R. Johnson is generalized to the operator case: for T ∈ (β(<b>H</b>), W(T<sup>n</sup>) ⊂ {Rez ≥ 0}, n = 1, 2,... if and only if T ≥ 0. Examples are given which show neither the necessity nor the sufficiency part of the theorem can be transplanted into the general Banach algebra setting. A containment result for the numerical range of a product is also proved.</p>https://thesis.library.caltech.edu/id/eprint/14076Indices of Principal Orders in Algebraic Number Fields
https://resolver.caltech.edu/CaltechTHESIS:08262021-155312274
Authors: Knight, Melvin John
Year: 1975
DOI: 10.7907/hcvt-yx83
<p>Let K be an extension of Q of degree n and D<sub>K</sub> the ring of integers of K. If θ is an algebraic integer of K and K = Q(θ), then Z[θ] is a suborder of D<sub>K</sub> of finite index. This index is called the index of θ. If k is a rational integer, the numbers θ and θ + k have equal indices. Define two numbers to be equivalent if their difference is a rational integer.</p>
<p>Using Schmidt's extension of Thue's Theorem it is shown that in any field of degree less than or equal to four there exist only a finite number of inequivalent numbers with index bounded by any given number. This is true for every finite extension of Q and a proof is given using a slight generalization of Schmidt's Theorem.</p>
<p>An application of Schmidt's Theorem to a problem on the units in a cyclic field of prime degree is given.</p>https://thesis.library.caltech.edu/id/eprint/14342The Matrix Equation F(A)X - XA = O
https://resolver.caltech.edu/CaltechTHESIS:04072017-144744158
Authors: Parker, Joseph A. Jr.
Year: 1976
DOI: 10.7907/X6W2-T505
<p>In this work, all matrices are assumed to have complex entries. The cases of
F(A) - XA = O where F(A) is a polynomial over C in A and
F(A) = (A<sup>*</sup>)<sup>-1</sup> are investigated. Canonical forms are derived for
solutions X to these equations. Other results are given for matrices
of the form A<sup>-1</sup>A<sup>*</sup>.</p>
<p>Let a set solutions {X<sub>i</sub>} be called a tower if X<sub>i+1</sub> = F(X<sub>i</sub>).
It is shown that towers occur for all nonsingular solutions of
(A<sup>*</sup>)<sup>-1</sup>X - XA = O if and only if A is normal. In contrast to this, there
is no polynomial for which only normal matrices A imply the existence of
towers for all solutions X of P(A)X - XA = O. On the other hand, conditions
are given for polynomials P, dependent upon spectrum of A, for
which only diagonalizable matrices A imply the existence of towers for all solutions
X of P(A)X - XA = O.</p>
https://thesis.library.caltech.edu/id/eprint/10133Applications of the Quaternions to the Study of Imaginary Quadratic Ring Class Groups
https://resolver.caltech.edu/CaltechETD:etd-04122004-144125
Authors: Hanlon, Philip James
Year: 1981
DOI: 10.7907/2G8A-Y604
<p>Let m = m<sub>1</sub>f<sup>2</sup> where m<sub>1</sub> 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 ℚ(√-m<sub>1</sub>). The proof given by Gauss involves the arithmetic of binary quadratic forms; Venkow (see [12]) obtained an alternative proof by embedding the field ℚ(√-m<sub>1</sub>) in the quaternion algebra over ℚ. 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.</p>
<p>Let H denote the quaternion algebra over ℚ, let E denote the maximal order in H and let U denote the group of 24 units in E. Let B<sub>1</sub>(m) be the set of quaternions in E with trace 0 and norm m. The group U acts on B<sub>0</sub>(m) by conjugation; let B<sub>1</sub>(m) denote the set of orbits of B<sub>0</sub>(m) under the action of U. For µ = ui<sub>1</sub> + vi<sub>2</sub> + wi<sub>3</sub>εB<sub>1</sub>(m) we let [u,v,w] denote the orbit containing µ.</p>
<p>Venkow proved Gauss's result by defining a sharply transitive action of Γ(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 ℚ(√-m<sub>1</sub>) and its prime divisor ℙ<sub>2</sub> is a regular ideal with respect to the discriminant -4m. It is shown that the class containing ℙ<sub>2</sub> maps [u,v,w] to [-u,-w,-v]. It is shown that if an ideal class ℂ maps [r,s,t] to [u,v,w] then the class ℂ<sup>-1</sup> maps [-r,-s,-t] to [-u,-v,-w]. From these two facts, several results follow. If ℂ maps [r,s,o] to [u,v,w] then ℂ has order 2 if one of u, v or w is 0. If ℂ maps [r,s,o] to [u,v,v] then ℂ has order 4 and the class ℂ<sup>2</sup> contains ℙ<sub>2</sub>. If ℂ maps [r,s,o] to [u,v,w] then ℂ<sup>-1</sup> maps [r,s,o] to [-u,-v,-w]. If m can be written as a sum of two squares then a class ℂ is the square of another class (i.e. ℂ is in the principal genus) if ℂ maps some bundle [u,v,w] to [-u,-v,-w].</p>
<p>We 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 ℚ(√-p). For n = 3 the number is given by</p>
<p>1/16 f(p)h(-4p) - 6h(-4p) + 2 if p ≡ 1 mod 4</p>
<p>1/8 f(p)h(-4p) - 6h(-4p) if p ≡ 3 mod 8</p>
<p>0 if p ≡ 7 mod 8</p>
<p>Here 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.</p>https://thesis.library.caltech.edu/id/eprint/1351