F(p'); GENERAL TYPES OF EQUATIONS IN A FINITE FIELD. superior and inferior limits for the number of solutions of the congruence
|
|
- Hubert Henry
- 6 years ago
- Views:
Transcription
1 26 A2ATHEMATICS: H. S. VANDIVER PROC. N. A. S. of A. Cayley. On the other -hand, W. R. Hamilton did not use the term group but favored the use of his own term quaternions for the development of a theory based upon the special abstract group of order S which-involves 6 operators of order 4. The associative law was later very commonly embodied in the definitions of an abstract group and the development of these groups did much to emphasize the fundamental importance of this law. These observations in regard to W. R. Hamilton seem to imply that the theory of abstract groups may reasonably be regarded as about ten years older than the assumption that A. Cayley was the founder of this theory would imply. Implicitly, W. R. Hamilton emphasized the importance of the quaternion group in his Lectures on Quaternions, page xxvi (1853). LIMITS FOR THE NUMBER OF SOLUTIONS OF CERTAIN GENERAL TYPES OF EQUATIONS IN A FINITE FIELD By H. S. VANDIVER DEPARTMENT OF APPLIED MATHEMATICS, UNIVERSITY OF TEXAS Communicated May 22, 1947 Dickson,I using cyclotomic integers, obtained an inferior limit for the number of solutions of the congruence xe + ye + ze 0 (mod p) (1) where e and p are given primes with xyz 0 (mod p). Hurwitz2 gave = superior and inferior limits for the number of solutions of the congruence axe + bye + czc O (mod p), (2) where abcxyz X 0 (mod p). Recently, the writer has considered generalizations of these results and has arrived at limits both superior and inferior, for the number of solutions of the equation ClXla + CXa C,Xs + cs+ = 0 (3) in the x's, where the a's are integers such that 0 < a. pt-1; s > 2; the c's belong to a finite field of order pt, p prime, which will be designated by and F(p'); C1C2... CsXlx2... Xs $ 0 in F(p'). As a corollary to this it is possible to show that if we take the c's in (3) as rational integers and put t = 1, so that we have a congruence modulo p in effect, then for p sufficiently large the congruence has solutions,
2 VOL. 33, 1947 MATHEMATICS: H. S. VANDIVER 237 with the x's all prime to p. There is a similar theorem involving algebraic numbers. However, in the present paper we shall confine ourselves to the derivation of certain quadratic relations between the numbers of solutions, xm, yi, of trinomial equations of the type 1 + axm = bytm (3a) for various values of a and b in F(pt), abxy 0, 'as given in Theorem I. Other quadratic relations connecting such solutions have been given before,3' 4 but those given here are of a quite different character, and they will be used in another paper in the proofs of the results concerning (3) referred to above. Consider a finite field of order pt designated by F(pt), where p is a prime. Write pt = 1 + mc. Let g be a generator of the cyclic group formed by the non-zero elements of F(pt). Further let (i, j) denote the number of solutions gt, gs of 1 + gi+rm = gj+s(3b) if r and s are each in the range 0, 1,..., c - 1, noting that gmc= 1. If we write ind. for index and gtndx = x, represent the index of (-1), modulo m, by e, then for any i and j it is known4 that (i,j) = (j+e,i+e) =(-j+e,i-j) = (i-j+e, -j) = (-i, j - i) = - i + -i+e). (4) Also we have4 (i, 0) = c-1; (i, j) = c, (4a) s i where i 0, 1,...,m- 1; j = 1, 2,...,m- 1 modulo m. We also note that (i, j) = (i + am, j + Om) for any integers a and 13. A fundamental relation we shall employ in what follows is (Mitchell,4 his 4I' function defined on p. 165; for b = a, a + b d) where 4a, d(af) a, d(al) = Pt (4b) "a, d( (a = (i, j)-ai+dj i,j a being a primitive mth root of unity, a and d any integers subject to the conditions a W 0, d W 0, a - d 0 0(mod m) with i and j ranging independently over the integers 0, 1,..., m - 1.
3 238 MATHEMATICS: H. S. VANDIVER PROC. N. A. 8. We shall now determine the quantities O to m-1 z (i, j) (i +. h, j + k) = Ahk (5) i, j First we determine Aoo. Now if a g 0 (mod m) d 0 0 (mod m), d - a 0 0 (mod m) then if a is a primitive mth root of unity, (4b).gives 0 to m_l Ei hijk (i,j) (i+h,j+k)ah+dk = t (6) Let pt = 1 + mc. Consider the summation m-1 Otom-1 m-1 E (i,j)(i + hi,j + k)c _ah+dk = y Cad (7) d=o hijk d=o For a 0 O(mod m), d 0 O(mod m), a - d g O(mod m) each term in the sum with respect to d = pt. Now consider the value when d = 0, which is 0 to m-1 E hijk (i,j)(i + h,j + k)a-ah Set i + h = i', j + k = j' then this expression becomes i,j,,$ This is obviously equal to Z (i, j)(i', jd)a W - i)a = Cao (8) if O to m-1 and each of these equals (-1) if a ) Oto m-1i ( 7 (ij)aia (i,j, /a-i'a (8a) 0(mod m), hence Cao=1 (9) Now consider the case when a - d =O(mod m) in (7). term reduces to or (i, j) (i,ji)aa(h-k) = Ca, E(i, j) a- ali -j)e(i,, y) C - a(i" If i - j = f, then for a 0 O(mod m), using (4), j,.f,(f + E -j)caaf = (f -j ) af - and similarly for the second factor. j,f C. = 1, a O(mod m). t The corresponding (10)
4 VOL. 33, 1947 MATHEMATICS: H. S. VANDIVER.239 Now if a 0=(mod m) in (8) then we find - Coo= (mc -1)2 (11) employing (8a) and (4a). We now simplify (7) under the assumption that a (7), (8), (9) and (10), we find 0 (mod m). Using d=0 Cad (m -2)p + 2; a O(mod m). (12) For the case where a = 0(mod m) we obtain in -the same way that (9) was derived Cw =1,d 0(modm). (13) rn-i Cod (mc-1)2 +m 1 (13a) d =0 Now the left hand member of (12) may be written d-o where Ahk is defined as in (5). Now rn-1 or from (12), hk AMhka) Ahka + d-o hk a.+- - Z Ahka ah(1±+ ak + + (rn1)k hk Oif k 0 (mod m). d-o ~ Ahak "ah+dk = iaia-ah hk me Ah,oaGk = (m -2)pt + 2, (14) h if a 0 (mod m) and it equals (mc-1)2 + n-l otherwise. mn-i mahoch = (m -2)(mc + 1)(m - 1) + 2(m - 1) + a=o (mc-1)2 + m-1
5 or A20ATHEMATICS: H. S. VANDIVER 240 PROC. N. A. S. or m2aoo = m2c2 + m3c - 3m2c + m2 Aoo =C(c12 + (m-l)c Now for a 0 (mod m) we have from (14) meaoa +a = asa((m - 2)pt + 2) h and for a 0 (mod m), mzah,= (mc-i)2+m-l. h (14a) rn-i m E EAha -ah+sa = (mc- 1)2 + m- 1- ((m -2)pt + 2) a=o h (C2 -C)M, or m2ato = m2(c2 -C) A,w = c2-c, h O (mod m). (15) We shall now show also that We have But by (1) Aok = c- c, k 0 0 (mod m). (16) Aok = (i, j)(i,j + k) ".7 (i, j) = (j+,i+ e) (i,j+k) = (j+ k + e,i+,e) j (i, j) (i, j + k) = Z i, j7j+e, i+e ((j + e) + k, i + e) (j + e, i + e), where j + e and i + e range over the same set 0, 1, , modulo m as do i and j so that this equation gives (16), using (15). We lastly determine Ahk with h 0, k 0 0 (mod m). Consider the sum m-1 Oto m-1 rn Z (ij)(i + hj, + k)cjah+dkvd = (17) d =O hijk We know from (6) that each term corresponding to a given d equals p'a-vd except when d 0, a 0, or d _ a, (mod m). For d- 0, then
6 VOL. 33, 1947 MATHEMATICS: H. S. VANDIVER 241 the corresponding term equals unity if a 0 (mod ni), using (9). For d = a we have by (10) that the term is a-a. B = 1 + -Vpt + a-2v pi a-(mr-l)v pt + -av _ -av pt Bd= 1 pt + a,(1 _ pt) (18) But Bad, can also be written as rn-i MAhva ah + E Ahka ah(l + ak-v + + a((m-1)(k-v)) k =O k#v and the last summation is 0. by (18) mahvaa-ah = 1 - pt + -av(l - pt) or and Ah,aah = -_-C a avc (19) Ah,a = cas- Ca-av+sa (20) We also have, from (17), for a = 0 (mod m) Bod, = Mc2-2c (21) for if we take (17) with a =- 0 (mod m) then for d 0 (mod m), we find that the corresponding term is (mc - 1)2 while the other terms, corresponding to each d are a-v, a- 2v...a (m-1)v, using (13). From (20) and (21) we find m-1m- a=o = - -h+s cs2 Aaah+sa (_caa Ca +s) + M2 2C MC2, a=o if v W s (mod m), or As,= c2; s W O, v 00, s - For v v (mod m). (21a) = s we have from the above A8S = c2- c; s 0 0 (mod m). (22). * we have, employing (14a), (15), (16), (21a) and (22), THEOREM I. Set 0 to m-1 Ahk = (i,j)(i + h,j + k), i,j where (i, j) is the number of -sets of values f and I in the set 0, 1,..., c - 1, which satisfy the equation
7 242 MATHEMATICS: T. Y. THOMAS PROC. N. A. S. 1 + gi+fm=g + m in the finite field F(pt) where p is a prime such that pl = 1 + cm, g being a primitive root in F(pt). Then Aoo = (c-1)2 + c(m-1); (23) Aho = Aok = c2 c, h O, k O (mod m); (24) Ahk = c2, with h 0 k (mod m); h 0 0, k 0 0 (mod m); (25) Ahk = C2 - C, (26) if h-k (mod m). 1 Crelle, 135, (1909). - Cf. also Pellet, Bull. Math. Soc. France, 15, (1886). 2 Crelle, 136, (1909). 3 THESE PROCEEDINGS, 32, (1946). 4 Mitchell, Proc. Amer. Math. Soc., 17, 167 (1916). CHARACTERISTIC COORDINA TES FOR HYPERBOLIC DIFFERENTIAL EQUATIONS IN THE LARGE By T. Y. THOMAS DEPARTMENT OF MATHEMATICS, INDIANA UNIVERSITY Communicated May 31, 1947 Partial differential equations of hyperbolic type are habitually treated by the introduction of characteristic coordinates. When such equations occur in physical problems their solutions in the large are necessarily demanded. In spite of this fact the existence of characteristic coordinates appears to be established only locally. We shall here give conditions for the existence of a (1, 1) differentiable transformation defining the characteristic coordinate system in the large. The form of these conditions is suitable for practical application but such applications of the theorem will not be included in this communication. Let D be an open simply connected two dimensional domain, finite or infinite, referred to a system of rectangular coordinates xa. At each point of D characteristic directions X are defined as the solutions of the equation = 0 in which the summation is over the values 1, 2 of the indices and it is supposed that the coefficients gag3 are continuous and have continuous first partial derivatives in D. Assuming det. lg,df < 0 (hyperbolic case) there will be exactly two distinct characteristic directions at each point of D and these directions will generate two families or congruences of characteristic curves each of which will, cover D completely. In fact-any point P of D is contained in a local co6rdinate system, e.g., a
xp = 9- ZP, yp + z y (mod p), (2) SUMMARY OF RESULTS AND PROOFS CONCERNING FERMA T'S DZPARTMZNT OF PURi MATHZMATICS, UNIVZRSITY of TExAs
VOL. 12, 1926 MA THEMA TICS: H. S. VANDI VER 767 gether in a subcontinuum of M. A point set M is strongly connected im kleinen if for every point P of M and for every positive number e there exists a positive
More informationELEMENTARY LINEAR ALGEBRA
ELEMENTARY LINEAR ALGEBRA K R MATTHEWS DEPARTMENT OF MATHEMATICS UNIVERSITY OF QUEENSLAND First Printing, 99 Chapter LINEAR EQUATIONS Introduction to linear equations A linear equation in n unknowns x,
More informationComputer Investigation of Difference Sets
Computer Investigation of Difference Sets By Harry S. Hayashi 1. Introduction. By a difference set of order k and multiplicity X is meant a set of k distinct residues n,r2,,rk (mod v) such that the congruence
More informationa 11 x 1 + a 12 x a 1n x n = b 1 a 21 x 1 + a 22 x a 2n x n = b 2.
Chapter 1 LINEAR EQUATIONS 11 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,, a n, b are given real
More informationS. Lie has thrown much new light on this operation. The assumption
600 MATHEMATICS: A. E. ROSS PRoc. N. A. S. The operation of finding the limit of an infinite series has been one of the most fruitful operations of all mathematics. While this is not a group operation
More informationELEMENTARY LINEAR ALGEBRA
ELEMENTARY LINEAR ALGEBRA K. R. MATTHEWS DEPARTMENT OF MATHEMATICS UNIVERSITY OF QUEENSLAND Second Online Version, December 1998 Comments to the author at krm@maths.uq.edu.au Contents 1 LINEAR EQUATIONS
More informationGroups. 3.1 Definition of a Group. Introduction. Definition 3.1 Group
C H A P T E R t h r e E Groups Introduction Some of the standard topics in elementary group theory are treated in this chapter: subgroups, cyclic groups, isomorphisms, and homomorphisms. In the development
More informationMATH 433 Applied Algebra Lecture 4: Modular arithmetic (continued). Linear congruences.
MATH 433 Applied Algebra Lecture 4: Modular arithmetic (continued). Linear congruences. Congruences Let n be a postive integer. The integers a and b are called congruent modulo n if they have the same
More informationAlgebraic structures I
MTH5100 Assignment 1-10 Algebraic structures I For handing in on various dates January March 2011 1 FUNCTIONS. Say which of the following rules successfully define functions, giving reasons. For each one
More informationELEMENTARY LINEAR ALGEBRA
ELEMENTARY LINEAR ALGEBRA K. R. MATTHEWS DEPARTMENT OF MATHEMATICS UNIVERSITY OF QUEENSLAND Corrected Version, 7th April 013 Comments to the author at keithmatt@gmail.com Chapter 1 LINEAR EQUATIONS 1.1
More informationAlgorithms lecture notes 1. Hashing, and Universal Hash functions
Algorithms lecture notes 1 Hashing, and Universal Hash functions Algorithms lecture notes 2 Can we maintain a dictionary with O(1) per operation? Not in the deterministic sense. But in expectation, yes.
More informationMATH Topics in Applied Mathematics Lecture 2-6: Isomorphism. Linear independence (revisited).
MATH 311-504 Topics in Applied Mathematics Lecture 2-6: Isomorphism. Linear independence (revisited). Definition. A mapping f : V 1 V 2 is one-to-one if it maps different elements from V 1 to different
More informationTHE MODULAR CURVE X O (169) AND RATIONAL ISOGENY
THE MODULAR CURVE X O (169) AND RATIONAL ISOGENY M. A. KENKU 1. Introduction Let N be an integer ^ 1. The affine modular curve Y 0 (N) parametrizes isomorphism classes of pairs (E ; C N ) where E is an
More informationMATH 423 Linear Algebra II Lecture 10: Inverse matrix. Change of coordinates.
MATH 423 Linear Algebra II Lecture 10: Inverse matrix. Change of coordinates. Let V be a vector space and α = [v 1,...,v n ] be an ordered basis for V. Theorem 1 The coordinate mapping C : V F n given
More informationTHEOREMS ON QUADRA TIC PARTITIONS. 5. An2-polyhedra. Let 7rr(K) = 0 for r = 1,..., n - 1, where n > 2.
60 MATHEMATICS: A. L. WHITEMAN PROC. N. A. S. 5. An2-polyhedra. Let 7rr(K) = 0 for r = 1,..., n - 1, where n > 2. In this case we may identify Fn+1 with'0 Hn(2) and bn+2 determines a homomorphism, (2)
More information1 4 which satisfies (2) identically in u for at least one value of the complex variable z then s c g.l.b. I ~m-1 y~~z cn, lar m- = 0 ( co < r, s < oo)
Nieuw Archief voor Wiskunde (3) III 13--19 (1955) FUNCTIONS WHICH ARE SYMMETRIC ABOUT SEVERAL POINTS BY PAUL ERDÖS and MICHAEL GOLOMB (Notre Dame University) (Purdue University) 1. Let 1(t) be a real-valued
More informationELEMENTARY LINEAR ALGEBRA
ELEMENTARY LINEAR ALGEBRA K R MATTHEWS DEPARTMENT OF MATHEMATICS UNIVERSITY OF QUEENSLAND Second Online Version, December 998 Comments to the author at krm@mathsuqeduau All contents copyright c 99 Keith
More informationTHE LIND-LEHMER CONSTANT FOR 3-GROUPS. Stian Clem 1 Cornell University, Ithaca, New York
#A40 INTEGERS 18 2018) THE LIND-LEHMER CONSTANT FOR 3-GROUPS Stian Clem 1 Cornell University, Ithaca, New York sac369@cornell.edu Christopher Pinner Department of Mathematics, Kansas State University,
More informationSpectra of Semidirect Products of Cyclic Groups
Spectra of Semidirect Products of Cyclic Groups Nathan Fox 1 University of Minnesota-Twin Cities Abstract The spectrum of a graph is the set of eigenvalues of its adjacency matrix A group, together with
More informationGroup, Rings, and Fields Rahul Pandharipande. I. Sets Let S be a set. The Cartesian product S S is the set of ordered pairs of elements of S,
Group, Rings, and Fields Rahul Pandharipande I. Sets Let S be a set. The Cartesian product S S is the set of ordered pairs of elements of S, A binary operation φ is a function, S S = {(x, y) x, y S}. φ
More informationMathematics Course 111: Algebra I Part I: Algebraic Structures, Sets and Permutations
Mathematics Course 111: Algebra I Part I: Algebraic Structures, Sets and Permutations D. R. Wilkins Academic Year 1996-7 1 Number Systems and Matrix Algebra Integers The whole numbers 0, ±1, ±2, ±3, ±4,...
More informationFINITE GROUPS AND EQUATIONS OVER FINITE FIELDS A PROBLEM SET FOR ARIZONA WINTER SCHOOL 2016
FINITE GROUPS AND EQUATIONS OVER FINITE FIELDS A PROBLEM SET FOR ARIZONA WINTER SCHOOL 2016 PREPARED BY SHABNAM AKHTARI Introduction and Notations The problems in Part I are related to Andrew Sutherland
More informationOn The Weights of Binary Irreducible Cyclic Codes
On The Weights of Binary Irreducible Cyclic Codes Yves Aubry and Philippe Langevin Université du Sud Toulon-Var, Laboratoire GRIM F-83270 La Garde, France, {langevin,yaubry}@univ-tln.fr, WWW home page:
More informationOn Quasi Quadratic Functionals and Existence of Related Sesquilinear Functionals
International Mathematical Forum, 2, 2007, no. 63, 3115-3123 On Quasi Quadratic Functionals and Existence of Related Sesquilinear Functionals Mehmet Açıkgöz University of Gaziantep, Faculty of Science
More informationFinite dihedral group algebras and coding theory
Finite dihedral group algebras and coding theory César Polcino Milies Universidade de São Paulo Basic FActs The basic elements to build a code are the following: Basic FActs The basic elements to build
More informationAugust 2015 Qualifying Examination Solutions
August 2015 Qualifying Examination Solutions If you have any difficulty with the wording of the following problems please contact the supervisor immediately. All persons responsible for these problems,
More informationARKIV FOR MATEMATIK Band 3 nr 24. Communicated 26 October 1955 by T. NAGELL. A theorem concerning the least quadratic residue and.
ARKIV FOR MATEMATIK Band 3 nr 24 Communicated 26 October 1955 by T. NAGELL A theorem concerning the least quadratic residue and non-residue By LARs FJELLSTEDT The purpose of this paper is to prove the
More informationE-SYMMETRIC NUMBERS (PUBLISHED: COLLOQ. MATH., 103(2005), NO. 1, )
E-SYMMETRIC UMBERS PUBLISHED: COLLOQ. MATH., 032005), O., 7 25.) GAG YU Abstract A positive integer n is called E-symmetric if there exists a positive integer m such that m n = φm), φn)). n is called E-asymmetric
More informationON DIVISION ALGEBRAS*
ON DIVISION ALGEBRAS* BY J. H. M. WEDDERBURN 1. The object of this paper is to develop some of the simpler properties of division algebras, that is to say, linear associative algebras in which division
More informationexceptional value turns out to be, "RAC" CURVATURE calling attention especially to the fundamental Theorem III, and to emphasize
VOL. 18, 1932 V,MA THEMA TICS: E. KASNER 267 COMPLEX GEOMETRY AND RELATIVITY: "RAC" CURVATURE BY EDWARD KASNER DEPARTMENT OF MATHEMATICS, COLUMBIA UNIVERSITY Communicated February 3, 1932 THEORY OF THE
More informationPROBLEMS ON CONGRUENCES AND DIVISIBILITY
PROBLEMS ON CONGRUENCES AND DIVISIBILITY 1. Do there exist 1,000,000 consecutive integers each of which contains a repeated prime factor? 2. A positive integer n is powerful if for every prime p dividing
More informationHOMEWORK Graduate Abstract Algebra I May 2, 2004
Math 5331 Sec 121 Spring 2004, UT Arlington HOMEWORK Graduate Abstract Algebra I May 2, 2004 The required text is Algebra, by Thomas W. Hungerford, Graduate Texts in Mathematics, Vol 73, Springer. (it
More informationSUM-PRODUCT ESTIMATES APPLIED TO WARING S PROBLEM MOD P
SUM-PRODUCT ESTIMATES APPLIED TO WARING S PROBLEM MOD P TODD COCHRANE AND CHRISTOPHER PINNER Abstract. Let γ(k, p) denote Waring s number (mod p) and δ(k, p) denote the ± Waring s number (mod p). We use
More informationp-regular functions and congruences for Bernoulli and Euler numbers
p-regular functions and congruences for Bernoulli and Euler numbers Zhi-Hong Sun( Huaiyin Normal University Huaian, Jiangsu 223001, PR China http://www.hytc.edu.cn/xsjl/szh Notation: Z the set of integers,
More informationGroups Subgroups Normal subgroups Quotient groups Homomorphisms Cyclic groups Permutation groups Cayley s theorem Class equations Sylow theorems
Group Theory Groups Subgroups Normal subgroups Quotient groups Homomorphisms Cyclic groups Permutation groups Cayley s theorem Class equations Sylow theorems Groups Definition : A non-empty set ( G,*)
More informationMetacommutation of Hurwitz primes
Metacommutation of Hurwitz primes Abhinav Kumar MIT Joint work with Henry Cohn January 10, 2013 Quaternions and Hurwitz integers Recall the skew-field of real quaternions H = R+Ri +Rj +Rk, with i 2 = j
More information120A LECTURE OUTLINES
120A LECTURE OUTLINES RUI WANG CONTENTS 1. Lecture 1. Introduction 1 2 1.1. An algebraic object to study 2 1.2. Group 2 1.3. Isomorphic binary operations 2 2. Lecture 2. Introduction 2 3 2.1. The multiplication
More informationNew Negative Latin Square Type Partial Difference Sets in Nonelementary Abelian 2-groups and 3-groups
New Negative Latin Square Type Partial Difference Sets in Nonelementary Abelian 2-groups and 3-groups John Polhill Department of Mathematics, Computer Science, and Statistics Bloomsburg University Bloomsburg,
More informationR. Popovych (Nat. Univ. Lviv Polytechnic )
UDC 512.624 R. Popovych (Nat. Univ. Lviv Polytechnic ) SHARPENING OF THE EXPLICIT LOWER BOUNDS ON THE ORDER OF ELEMENTS IN FINITE FIELD EXTENSIONS BASED ON CYCLOTOMIC POLYNOMIALS ПІДСИЛЕННЯ ЯВНИХ НИЖНІХ
More informationMath 120. Groups and Rings Midterm Exam (November 8, 2017) 2 Hours
Math 120. Groups and Rings Midterm Exam (November 8, 2017) 2 Hours Name: Please read the questions carefully. You will not be given partial credit on the basis of having misunderstood a question, and please
More informationSOME AMAZING PROPERTIES OF THE FUNCTION f(x) = x 2 * David M. Goldschmidt University of California, Berkeley U.S.A.
SOME AMAZING PROPERTIES OF THE FUNCTION f(x) = x 2 * David M. Goldschmidt University of California, Berkeley U.S.A. 1. Introduction Today we are going to have a look at one of the simplest functions in
More informationINTRODUCTION TO THE GROUP THEORY
Lecture Notes on Structure of Algebra INTRODUCTION TO THE GROUP THEORY By : Drs. Antonius Cahya Prihandoko, M.App.Sc e-mail: antoniuscp.fkip@unej.ac.id Mathematics Education Study Program Faculty of Teacher
More informationarxiv: v1 [math.nt] 4 Jun 2018
On the orbit of a post-critically finite polynomial of the form x d +c arxiv:1806.01208v1 [math.nt] 4 Jun 2018 Vefa Goksel Department of Mathematics University of Wisconsin Madison, WI 53706, USA E-mail:
More informationAttempt QUESTIONS 1 and 2, and THREE other questions. penalised if you attempt additional questions.
UNIVERSITY OF EAST ANGLIA School of Mathematics Main Series UG Examination 2017 18 FERMAT S LAST THEOREM MTHD6024B Time allowed: 3 Hours Attempt QUESTIONS 1 and 2, and THREE other questions. penalised
More informationEXERCISES. a b = a + b l aq b = ab - (a + b) + 2. a b = a + b + 1 n0i) = oii + ii + fi. A. Examples of Rings. C. Ring of 2 x 2 Matrices
/ rings definitions and elementary properties 171 EXERCISES A. Examples of Rings In each of the following, a set A with operations of addition and multiplication is given. Prove that A satisfies all the
More informationON THE THEORY OF ASSOCIATIVE DIVISION ALGEBRAS*
ON THE THEORY OF ASSOCIATIVE DIVISION ALGEBRAS* BY OLIVE C. HAZLETT 1. Relation to the literature. There is a famous theorem to the effect that the only linear associative algebras over the field of all
More informationON THE SEMIPRIMITIVITY OF CYCLIC CODES
ON THE SEMIPRIMITIVITY OF CYCLIC CODES YVES AUBRY AND PHILIPPE LANGEVIN Abstract. We prove, without assuming the Generalized Riemann Hypothesis, but with at most one exception, that an irreducible cyclic
More informationM381 Number Theory 2004 Page 1
M81 Number Theory 2004 Page 1 [[ Comments are written like this. Please send me (dave@wildd.freeserve.co.uk) details of any errors you find or suggestions for improvements. ]] Question 1 20 = 2 * 10 +
More informationDiophantine Equations and Hilbert s Theorem 90
Diophantine Equations and Hilbert s Theorem 90 By Shin-ichi Katayama Department of Mathematical Sciences, Faculty of Integrated Arts and Sciences The University of Tokushima, Minamijosanjima-cho 1-1, Tokushima
More informationNewton, Fermat, and Exactly Realizable Sequences
1 2 3 47 6 23 11 Journal of Integer Sequences, Vol. 8 (2005), Article 05.1.2 Newton, Fermat, and Exactly Realizable Sequences Bau-Sen Du Institute of Mathematics Academia Sinica Taipei 115 TAIWAN mabsdu@sinica.edu.tw
More informationMATH 361: NUMBER THEORY FOURTH LECTURE
MATH 361: NUMBER THEORY FOURTH LECTURE 1. Introduction Everybody knows that three hours after 10:00, the time is 1:00. That is, everybody is familiar with modular arithmetic, the usual arithmetic of the
More informationA SURVEY OF DIFFERENCE SETS1
A SURVEY OF DIFFERENCE SETS1 MARSHALL HALL, JR. 1. Introduction. A set of k distinct residues dx, d2,,dk modulo v is called a difference set D if every residue >^0 (mod v) can be expressed in exactly X
More informationA connection between number theory and linear algebra
A connection between number theory and linear algebra Mark Steinberger Contents 1. Some basics 1 2. Rational canonical form 2 3. Prime factorization in F[x] 4 4. Units and order 5 5. Finite fields 7 6.
More informationIntegral Similarity and Commutators of Integral Matrices Thomas J. Laffey and Robert Reams (University College Dublin, Ireland) Abstract
Integral Similarity and Commutators of Integral Matrices Thomas J. Laffey and Robert Reams (University College Dublin, Ireland) Abstract Let F be a field, M n (F ) the algebra of n n matrices over F and
More informationSummary Slides for MATH 342 June 25, 2018
Summary Slides for MATH 342 June 25, 2018 Summary slides based on Elementary Number Theory and its applications by Kenneth Rosen and The Theory of Numbers by Ivan Niven, Herbert Zuckerman, and Hugh Montgomery.
More informationNUMBER SYSTEMS. Number theory is the study of the integers. We denote the set of integers by Z:
NUMBER SYSTEMS Number theory is the study of the integers. We denote the set of integers by Z: Z = {..., 3, 2, 1, 0, 1, 2, 3,... }. The integers have two operations defined on them, addition and multiplication,
More informationIMPROVEMENT OF A THEOREM OF LINNIK AND WALFISZ 423 IMPROVEMENT OF A THEOREM OF LINNIK AND WALFISZ. By S. CHOWLA
IMPROVEMENT OF A THEOREM OF LINNIK AND WALFISZ 423 IMPROVEMENT OF A THEOREM OF LINNIK AND WALFISZ By S. CHOWLA [Received 19 November 1946 Read 19 December 1946] 1. On the basis of the hitherto unproved
More informationChapter 9 Factorisation and Discrete Logarithms Using a Factor Base
Chapter 9 Factorisation and Discrete Logarithms Using a Factor Base February 15, 2010 9 The two intractable problems which are at the heart of public key cryptosystems, are the infeasibility of factorising
More informationChapter 2. Linear Algebra. rather simple and learning them will eventually allow us to explain the strange results of
Chapter 2 Linear Algebra In this chapter, we study the formal structure that provides the background for quantum mechanics. The basic ideas of the mathematical machinery, linear algebra, are rather simple
More informationSUBSPACE LATTICES OF FINITE VECTOR SPACES ARE 5-GENERATED
SUBSPACE LATTICES OF FINITE VECTOR SPACES ARE 5-GENERATED LÁSZLÓ ZÁDORI To the memory of András Huhn Abstract. Let n 3. From the description of subdirectly irreducible complemented Arguesian lattices with
More informationA GENERALIZATION OF BI IDEALS IN SEMIRINGS
BULLETIN OF THE INTERNATIONAL MATHEMATICAL VIRTUAL INSTITUTE ISSN (p) 2303-4874, ISSN (o) 2303-4955 www.imvibl.org /JOURNALS / BULLETIN Vol. 8(2018), 123-133 DOI: 10.7251/BIMVI1801123M Former BULLETIN
More informationbriefly consider in connection with the differential equations of the type G catenaries, (5) the co I
VOL. 28, 1942 MA THEMA TICS: E. KA SNER 333 Kasner, "Transformation Theory of Isothermal Families and Certain Related Trajectories," Revista de Matematica, 2, 17-24 (1941). De Cicco, "The Two Conformal
More informationAMBIGUOUS FORMS AND IDEALS IN QUADRATIC ORDERS. Copyright 2009 Please direct comments, corrections, or questions to
AMBIGUOUS FORMS AND IDEALS IN QUADRATIC ORDERS JOHN ROBERTSON Copyright 2009 Please direct comments, corrections, or questions to jpr2718@gmail.com This note discusses the possible numbers of ambiguous
More informationALGEBRAS AND THEIR ARITHMETICS*
1924.] ALGEBRAS AND THEIR ARITHMETICS 247 ALGEBRAS AND THEIR ARITHMETICS* BY L. E. DICKSON 1. Introduction. Beginning with Hamilton's discovery of quaternions eighty years ago, there has been a widespread
More informationSums of Semiprime, Z, and D L-Ideals in a Class of F-Rings
Digital Commons@ Loyola Marymount University and Loyola Law School Mathematics Faculty Works Mathematics 8-1-1990 Sums of Semiprime, Z, and D L-Ideals in a Class of F-Rings Suzanne Larson Loyola Marymount
More informationNOTES ON HYPERBOLICITY CONES
NOTES ON HYPERBOLICITY CONES Petter Brändén (Stockholm) pbranden@math.su.se Berkeley, October 2010 1. Hyperbolic programming A hyperbolic program is an optimization problem of the form minimize c T x such
More informationUNIVERSITY OF NORTH CAROLINA Department of Statistics Chapel Hill, N. C.
UNIVERSITY OF NORTH CAROLINA Department of Statistics Chapel Hill, N. C. Mathematical Sciences Directorate Air Force Office of Scientific Research Washington 25, D. C. AFOSR Report No. THEOREMS IN THE
More information3+4=2 5+6=3 7 4=4. a + b =(a + b) mod m
Rings and fields The ring Z m -part2(z 5 and Z 8 examples) Suppose we are working in the ring Z 5, consisting of the set of congruence classes Z 5 := {[0] 5, [1] 5, [2] 5, [3] 5, [4] 5 } with the operations
More informationCOURSE SUMMARY FOR MATH 504, FALL QUARTER : MODERN ALGEBRA
COURSE SUMMARY FOR MATH 504, FALL QUARTER 2017-8: MODERN ALGEBRA JAROD ALPER Week 1, Sept 27, 29: Introduction to Groups Lecture 1: Introduction to groups. Defined a group and discussed basic properties
More informationON STRONGLY PRIME IDEALS AND STRONGLY ZERO-DIMENSIONAL RINGS. Christian Gottlieb
ON STRONGLY PRIME IDEALS AND STRONGLY ZERO-DIMENSIONAL RINGS Christian Gottlieb Department of Mathematics, University of Stockholm SE-106 91 Stockholm, Sweden gottlieb@math.su.se Abstract A prime ideal
More informationand A", where (1) p- 8/+ 1 = X2 + Y2 = C2 + 2D2, C = X=\ (mod4).
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 87, Number 3. March 1983 THE Odie PERIOD POLYNOMIAL RONALD J. EVANS1 Abstract. The coefficients and the discriminant of the octic period polynomial
More informationCongruence of Integers
Congruence of Integers November 14, 2013 Week 11-12 1 Congruence of Integers Definition 1. Let m be a positive integer. For integers a and b, if m divides b a, we say that a is congruent to b modulo m,
More informationAnother Proof of Nathanson s Theorems
1 2 3 47 6 23 11 Journal of Integer Sequences, Vol. 14 (2011), Article 11.8.4 Another Proof of Nathanson s Theorems Quan-Hui Yang School of Mathematical Sciences Nanjing Normal University Nanjing 210046
More informationarxiv:math/ v3 [math.gr] 21 Feb 2006
arxiv:math/0407446v3 [math.gr] 21 Feb 2006 Separability of Solvable Subgroups in Linear Groups Roger Alperin and Benson Farb August 11, 2004 Abstract Let Γ be a finitely generated linear group over a field
More informationFIXED-POINT FREE ENDOMORPHISMS OF GROUPS RELATED TO FINITE FIELDS
FIXED-POINT FREE ENDOMORPHISMS OF GROUPS RELATED TO FINITE FIELDS LINDSAY N. CHILDS Abstract. Let G = F q β be the semidirect product of the additive group of the field of q = p n elements and the cyclic
More informationarxiv: v2 [math.nt] 4 Jun 2016
ON THE p-adic VALUATION OF STIRLING NUMBERS OF THE FIRST KIND PAOLO LEONETTI AND CARLO SANNA arxiv:605.07424v2 [math.nt] 4 Jun 206 Abstract. For all integers n k, define H(n, k) := /(i i k ), where the
More informationThe Distribution of Generalized Sum-of-Digits Functions in Residue Classes
Journal of umber Theory 79, 9426 (999) Article ID jnth.999.2424, available online at httpwww.idealibrary.com on The Distribution of Generalized Sum-of-Digits Functions in Residue Classes Abigail Hoit Department
More informationConjugacy classes of torsion in GL_n(Z)
Electronic Journal of Linear Algebra Volume 30 Volume 30 (2015) Article 32 2015 Conjugacy classes of torsion in GL_n(Z) Qingjie Yang Renmin University of China yangqj@ruceducn Follow this and additional
More informationCyclotomic Resultants. By D. H. Lehmer and Emma Lehmer. Dedicated to Daniel Shanks on his 10 th birthday
mathematics of computation volume 48. number 177 january 1w7. paces 211-216 Cyclotomic Resultants By D. H. Lehmer and Emma Lehmer Dedicated to Daniel Shanks on his 10 th birthday Abstract. This paper examines
More information= i 0. a i q i. (1 aq i ).
SIEVED PARTITIO FUCTIOS AD Q-BIOMIAL COEFFICIETS Fran Garvan* and Dennis Stanton** Abstract. The q-binomial coefficient is a polynomial in q. Given an integer t and a residue class r modulo t, a sieved
More informationNUMERICAL MONOIDS (I)
Seminar Series in Mathematics: Algebra 2003, 1 8 NUMERICAL MONOIDS (I) Introduction The numerical monoids are simple to define and naturally appear in various parts of mathematics, e.g. as the values monoids
More informationMINIMAL GENERATING SETS OF GROUPS, RINGS, AND FIELDS
MINIMAL GENERATING SETS OF GROUPS, RINGS, AND FIELDS LORENZ HALBEISEN, MARTIN HAMILTON, AND PAVEL RŮŽIČKA Abstract. A subset X of a group (or a ring, or a field) is called generating, if the smallest subgroup
More informationIntroduction to Lucas Sequences
A talk given at Liaoning Normal Univ. (Dec. 14, 017) Introduction to Lucas Sequences Zhi-Wei Sun Nanjing University Nanjing 10093, P. R. China zwsun@nju.edu.cn http://math.nju.edu.cn/ zwsun Dec. 14, 017
More informationGRE Subject test preparation Spring 2016 Topic: Abstract Algebra, Linear Algebra, Number Theory.
GRE Subject test preparation Spring 2016 Topic: Abstract Algebra, Linear Algebra, Number Theory. Linear Algebra Standard matrix manipulation to compute the kernel, intersection of subspaces, column spaces,
More informationMULTIPLICATIVE SUBGROUPS OF FINITE INDEX IN A DIVISION RING GERHARD TURNWALD. (Communicated by Maurice Auslander)
PROCEEDINGS of the AMERICAN MATHEMATICAL SOCIETY Volume 120, Number 2, February 1994 MULTIPLICATIVE SUBGROUPS OF FINITE INDEX IN A DIVISION RING GERHARD TURNWALD (Communicated by Maurice Auslander) Abstract.
More informationTHE NUMBER OF DIOPHANTINE QUINTUPLES. Yasutsugu Fujita College of Industrial Technology, Nihon University, Japan
GLASNIK MATEMATIČKI Vol. 45(65)(010), 15 9 THE NUMBER OF DIOPHANTINE QUINTUPLES Yasutsugu Fujita College of Industrial Technology, Nihon University, Japan Abstract. A set a 1,..., a m} of m distinct positive
More informationGENERATING SETS KEITH CONRAD
GENERATING SETS KEITH CONRAD 1 Introduction In R n, every vector can be written as a unique linear combination of the standard basis e 1,, e n A notion weaker than a basis is a spanning set: a set of vectors
More informationQuadratics. Shawn Godin. Cairine Wilson S.S Orleans, ON October 14, 2017
Quadratics Shawn Godin Cairine Wilson S.S Orleans, ON Shawn.Godin@ocdsb.ca October 14, 2017 Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, 2017 1 / 110 Binary Quadratic Form A form is a homogeneous
More informationMath 312/ AMS 351 (Fall 17) Sample Questions for Final
Math 312/ AMS 351 (Fall 17) Sample Questions for Final 1. Solve the system of equations 2x 1 mod 3 x 2 mod 7 x 7 mod 8 First note that the inverse of 2 is 2 mod 3. Thus, the first equation becomes (multiply
More informationMATH 106 LINEAR ALGEBRA LECTURE NOTES
MATH 6 LINEAR ALGEBRA LECTURE NOTES FALL - These Lecture Notes are not in a final form being still subject of improvement Contents Systems of linear equations and matrices 5 Introduction to systems of
More informationAn Elementary Proof that any Natural Number can be Written as the Sum of Three Terms of the Sequence n2
1 47 6 11 Journal of Integer Sequences, Vol. 17 (014), Article 14.7.6 An Elementary Proof that any Natural Number can be Written as the Sum of Three Terms of the Sequence n Bakir Farhi Department of Mathematics
More informationGENERATING SETS KEITH CONRAD
GENERATING SETS KEITH CONRAD 1 Introduction In R n, every vector can be written as a unique linear combination of the standard basis e 1,, e n A notion weaker than a basis is a spanning set: a set of vectors
More informationAn infinite family of Goethals-Seidel arrays
An infinite family of Goethals-Seidel arrays Mingyuan Xia Department of Mathematics, Central China Normal University, Wuhan, Hubei 430079, China Email: xiamy@ccnu.edu.cn Tianbing Xia, Jennifer Seberry
More informationSimultaneous Diophantine Approximation with Excluded Primes. László Babai Daniel Štefankovič
Simultaneous Diophantine Approximation with Excluded Primes László Babai Daniel Štefankovič Dirichlet (1842) Simultaneous Diophantine Approximation Given reals integers α,,...,, 1 α 2 α n Q r1,..., r n
More informationDetection Whether a Monoid of the Form N n / M is Affine or Not
International Journal of Algebra Vol 10 2016 no 7 313-325 HIKARI Ltd wwwm-hikaricom http://dxdoiorg/1012988/ija20166637 Detection Whether a Monoid of the Form N n / M is Affine or Not Belgin Özer and Ece
More informationDifference Systems of Sets and Cyclotomy
Difference Systems of Sets and Cyclotomy Yukiyasu Mutoh a,1 a Graduate School of Information Science, Nagoya University, Nagoya, Aichi 464-8601, Japan, yukiyasu@jim.math.cm.is.nagoya-u.ac.jp Vladimir D.
More informationSOME VARIANTS OF LAGRANGE S FOUR SQUARES THEOREM
Acta Arith. 183(018), no. 4, 339 36. SOME VARIANTS OF LAGRANGE S FOUR SQUARES THEOREM YU-CHEN SUN AND ZHI-WEI SUN Abstract. Lagrange s four squares theorem is a classical theorem in number theory. Recently,
More informationOn Strongly Prime Semiring
BULLETIN of the Malaysian Mathematical Sciences Society http://math.usm.my/bulletin Bull. Malays. Math. Sci. Soc. (2) 30(2) (2007), 135 141 On Strongly Prime Semiring T.K. Dutta and M.L. Das Department
More informationMCS 563 Spring 2014 Analytic Symbolic Computation Monday 14 April. Binomial Ideals
Binomial Ideals Binomial ideals offer an interesting class of examples. Because they occur so frequently in various applications, the development methods for binomial ideals is justified. 1 Binomial Ideals
More information