Normal cyclotomic schemes over a Galois ring
|
|
- George Turner
- 5 years ago
- Views:
Transcription
1 St.Petersburg Department of Steklov Mathematical Institute Russia (mostly joint work with Ilia Ponomarenko) The 11th International Conference on Finite Fields Friday, July 26, Magdeburg
2 Outline Cyclotomic schemes over a field 1 Cyclotomic schemes over a field Definition Automorphism group 2 Finite commutative rings Normality 3 Main Theorem Sketch of proof
3 Definition (Delsarte, 1973) Definition Automorphism group Let F = F q be a finite field of order q > 1 and K a subgroup of index m in its multiplicative group F. For each a F set E a = {(x, y) F F : x y ak }. Under a cyclotomic scheme C = Cyc(K, F) over F we mean the pair (F, {E a : a F}). Any such C is an association scheme, rk(c) = 1 + m. Moreover, C is a Cayley scheme over the additive group of F. The intersection numbers of C are cyclotomic numbers. The explicit evaluation of them is a hard number-theoretic problem.
4 McConnel Theorem (1963) Definition Automorphism group For a cyclotomic scheme C = Cyc(K, F) set Aut(C) = {f Sym(F) : (E a ) f = E a, a F}. Denote by AΓL 1 (F) the group consisting of all 1-dimensional semi-affine transformations of F: AΓL 1 (F) = {x ax σ + b : a F, b F, σ Aut(F)}. Theorem If rk(c) > 2, then Aut(C) AΓL 1 (F). Note. If rk(c) = 2, then Aut(C) = Sym(F).
5 Finite commutative rings Normality Decompositions in a finite commutative ring Let R be a finite commutative ring with identity. Then R = i R i where R i is a local ring for all i. If R is a local ring, then R/ rad(r) =: F is a field. Moreover, R = T U where T is the Teichmüller group of R, and U is the group of its principal units. The group T is isomorphic to F whereas U = 1 + rad(r) is a p-group where p = char(f).
6 Galois rings Cyclotomic schemes over a field Finite commutative rings Normality Definition A finite local ring R is Galois if rad(r) = pr. All Galois rings with char(r) = p n and F = p d are isomorphic. Any of them is denoted by GR(p n, d). R + is a homocyclic p-group of exponent p n and rank d, I(R) = {p i R : 0 i n}, the quotient homomorphism Aut(R) Aut(F) is a bijection, R is additively generated by T, n 1 x = x i p i, x i T {0}. i=0 Examples: GR(p n, 1) = Z p n, GR(p, d) = F p d.
7 Finite commutative rings Normality Multiplicative group of a Galois ring Let C m denote a cyclic group of order m. Theorem If R = GR(p n, d), then R = T U where T = C p d 1 If p is odd, then U = (C p n 1) d, If p = 2, then { U 1, if n = 1 = C 2 C 2 n 2 (C 2 n 1) d 1, if n > 1. Example: If R = GR(2 n, 1) = Z 2 n, then R = 1 5.
8 Finite commutative rings Normality Cyclotomic schemes over a ring (Goldbach, Claasen) Let R be a finite commutative ring with identity and let K be a subgroup of index m in its multiplicative group R. For each a R set E a = {(x, y) R R : x y ak }. Under a cyclotomic scheme C = Cyc(K, R) over R we mean the pair (R, {E a : a R}). Any such C is an association scheme, rk(c) = 1 + m. Moreover, C is a Cayley scheme over the additive group of R.
9 Normal schemes Cyclotomic schemes over a field Finite commutative rings Normality Let C = Cyc(K, R) with K R be a cyclotomic scheme over a finite commutative ring R. Definition The scheme C is normal if Aut(C) AΓL 1 (R). McConnel s Theorem implies the following statement. Theorem Let R = F q be a field. Then the scheme C is normal if and only if rk(c) > 2 or q = 2, 3, 4. Proof. If rk(c) = 2, then Aut(C) = Sym(F q ), and consequently Aut(C) AΓL 1 (F q ) q = 2, 3, 4.
10 Reduction to the local case Finite commutative rings Normality Let R = i R i be a finite commutative ring and C = Cyc(K, R) a cyclotomic scheme over R where K R. For each i set C i = Cyc(K i, R i ) where K i = (ϕ i ) 1 (K ) with ϕ i the natural embedding of R i into R : x (1,..., x,..., 1). Theorem The scheme C is normal if and only if the scheme C i is normal for all i.
11 Main Theorem Sketch of proof Pure and quasipure subgroups of R Let R be a commutative ring with identity. For a group K R denote by I K the largest of all ideals I of R such that K + I = K (equivalently, 1 + I K ). Definition The group K is said to be pure if I K = 0; it is said to be quasipure if I K I 0 where I 0 = ann(rad(r)). If R is a Galois ring of characteristic p n, then I 0 = p n 1 R.
12 Statements Cyclotomic schemes over a field Main Theorem Sketch of proof Theorem Let R = GR(p n, d) be a Galois ring other than a field. Then the scheme Cyc(K, R) is normal if and only if the group K is pure for q > 2 and quasipure for q = 2. Corollary If R is not isomorphic to Z 2 n, then Cyc(K, R) is normal if and only if K is pure.
13 Necessity Cyclotomic schemes over a field Main Theorem Sketch of proof Let R be a local ring, C = Cyc(K, R) and I 0 = ann(rad(r)). Theorem Suppose that C is a normal scheme and K = K + I for some ideal I of R. Then I = 0 whenever q > 2. Moreover, if q = 2, then I I 0. Proof. Assuming I 0, one can show that C is the generalized wreath product of smaller schemes and consequently the group Aut(C) contains a subgroup which is not inside AΓL 1 (R) unless q = 2 and I I 0. Cyclotomic S-ring over G = R + corresponding to C: A = span{σx; X Orb(K, G)} ZG.
14 Sufficiency: q = 2 Main Theorem Sketch of proof If q = 2, then R = Z 2 n and I 0 = 2 n 1 R. Theorem Let R = Z 2 n and I K I 0. Then K belongs to one of the following families of groups: {1}, n 1, {1, 1}, n 2, {1, 2 n 1 + 1}, n 3, {1, 2 n 1 1}, n 3, {1, 2 n 1 1, 2 n 1 + 1, 1}, n 4, {1, 2 n 2 1, 2 n 1 + 1, 3 2 n 2 1}, n 4. Moreover, the scheme Cyc(K, R) is normal.
15 Sufficiency: q > 2 Main Theorem Sketch of proof All we need to prove is the following statement: if I K = 0, then the scheme Cyc(K, R) is normal. Let π : R R/I 0 be the quotient epimorphism. Theorem Let K R be a pure group, C = Cyc(K, R), and C = Cyc(K, R ) where K = π(k ) and R = π(r). Then the scheme C is normal whenever so is the scheme C. Theorem If R = GR(p n, d), then the group K is pure whenever n > 2 for an odd p and n > 3 for p = 2. Thus the proof of sufficiency is reduced to the following cases: n = 2, p is arbitrary and n = 3, p = 2.
16 Sufficiency Cyclotomic schemes over a field Main Theorem Sketch of proof It suffices to prove that Aut(C) 0,1 Aut(R) where Aut(C) 0,1 = {γ Aut(C) : 0 γ = 0, 1 γ = 1}. Lemma Let R be a local commutative ring. Suppose that a permutation γ Sym(R) fixes 0 and 1 and normalizes the group AGL 1 (R). Then γ Aut(R). Multiplication S-ring of the scheme C: by definition it equals the S-ring over the group R that corresponds to the Cayley scheme ((C 0 ) R ) R. In general this S-ring is not cyclotomic.
17 Papers Papers I R. McConnel. Pseudo-ordered polynomials over a finite field. Acta Arith. 8: , B. R. McDonald. Finite rings with identity. Pure and Applied Mathematics, Vol. 28, Marcel Dekker Inc., New York, S. Evdokimov, I. Ponomarenko. Normal cyclotomic schemes over a finite commutative ring. Algebra & Analysis 19(6):58 84, 2007.
Coset closure of a circulant S-ring and schurity problem
Coset closure of a circulant S-ring and schurity problem Ilya Ponomarenko St.Petersburg Department of V.A.Steklov Institute of Mathematics of the Russian Academy of Sciences Modern Trends in Algebraic
More informationSeparability of Schur rings and Cayley graph isomorphism problem
Separability of Schur rings and Cayley graph isomorphism problem Grigory Ryabov Novosibirsk State University Symmetry vs Regularity, Pilsen, July 1-7, 2018 1 / 14 S-rings G is a finite group, e is the
More informationButson-Hadamard matrices in association schemes of class 6 on Galois rings of characteristic 4
Butson-Hadamard matrices in association schemes of class 6 on Galois rings of characteristic 4 Akihiro Munemasa Tohoku University (joint work with Takuya Ikuta) November 17, 2017 Combinatorics Seminar
More informationButson-Hadamard matrices in association schemes of class 6 on Galois rings of characteristic 4
Butson-Hadamard matrices in association schemes of class 6 on Galois rings of characteristic 4 Akihiro Munemasa Tohoku University (joint work with Takuya Ikuta) November 17, 2017 Combinatorics Seminar
More information1. Group Theory Permutations.
1.1. Permutations. 1. Group Theory Problem 1.1. Let G be a subgroup of S n of index 2. Show that G = A n. Problem 1.2. Find two elements of S 7 that have the same order but are not conjugate. Let π S 7
More informationFields and Galois Theory. Below are some results dealing with fields, up to and including the fundamental theorem of Galois theory.
Fields and Galois Theory Below are some results dealing with fields, up to and including the fundamental theorem of Galois theory. This should be a reasonably logical ordering, so that a result here should
More informationTHE GROUP OF UNITS OF SOME FINITE LOCAL RINGS I
J Korean Math Soc 46 (009), No, pp 95 311 THE GROUP OF UNITS OF SOME FINITE LOCAL RINGS I Sung Sik Woo Abstract The purpose of this paper is to identify the group of units of finite local rings of the
More information7 Semidirect product. Notes 7 Autumn Definition and properties
MTHM024/MTH74U Group Theory Notes 7 Autumn 20 7 Semidirect product 7. Definition and properties Let A be a normal subgroup of the group G. A complement for A in G is a subgroup H of G satisfying HA = G;
More information3 Galois Theory. 3.1 Definitions and Examples
3 Galois Theory 3.1 Definitions and Examples This section of notes roughly follows Section 14.1 in Dummit and Foote. Let F be a field and let f (x) 2 F[x]. In the previous chapter, we proved that there
More informationAlgebra SEP Solutions
Algebra SEP Solutions 17 July 2017 1. (January 2017 problem 1) For example: (a) G = Z/4Z, N = Z/2Z. More generally, G = Z/p n Z, N = Z/pZ, p any prime number, n 2. Also G = Z, N = nz for any n 2, since
More informationAlgebra Exam Topics. Updated August 2017
Algebra Exam Topics Updated August 2017 Starting Fall 2017, the Masters Algebra Exam will have 14 questions. Of these students will answer the first 8 questions from Topics 1, 2, and 3. They then have
More informationAlgebra Homework, Edition 2 9 September 2010
Algebra Homework, Edition 2 9 September 2010 Problem 6. (1) Let I and J be ideals of a commutative ring R with I + J = R. Prove that IJ = I J. (2) Let I, J, and K be ideals of a principal ideal domain.
More informationMAIN THEOREM OF GALOIS THEORY
MAIN THEOREM OF GALOIS THEORY Theorem 1. [Main Theorem] Let L/K be a finite Galois extension. and (1) The group G = Gal(L/K) is a group of order [L : K]. (2) The maps defined by and f : {subgroups of G}!
More informationON GALOIS GROUPS OF ABELIAN EXTENSIONS OVER MAXIMAL CYCLOTOMIC FIELDS. Mamoru Asada. Introduction
ON GALOIS GROUPS OF ABELIAN ETENSIONS OVER MAIMAL CYCLOTOMIC FIELDS Mamoru Asada Introduction Let k 0 be a finite algebraic number field in a fixed algebraic closure Ω and ζ n denote a primitive n-th root
More informationDefinitions. Notations. Injective, Surjective and Bijective. Divides. Cartesian Product. Relations. Equivalence Relations
Page 1 Definitions Tuesday, May 8, 2018 12:23 AM Notations " " means "equals, by definition" the set of all real numbers the set of integers Denote a function from a set to a set by Denote the image of
More informationOn the linearity of HNN-extensions with abelian base groups
On the linearity of HNN-extensions with abelian base groups Dimitrios Varsos Joint work with V. Metaftsis and E. Raptis with base group K a polycyclic-by-finite group and associated subgroups A and B of
More informationHOMEWORK 3 LOUIS-PHILIPPE THIBAULT
HOMEWORK 3 LOUIS-PHILIPPE THIBAULT Problem 1 Let G be a group of order 56. We have that 56 = 2 3 7. Then, using Sylow s theorem, we have that the only possibilities for the number of Sylow-p subgroups
More informationTC10 / 3. Finite fields S. Xambó
TC10 / 3. Finite fields S. Xambó The ring Construction of finite fields The Frobenius automorphism Splitting field of a polynomial Structure of the multiplicative group of a finite field Structure of the
More informationA modular absolute bound condition for primitive association schemes
J Algebr Comb (2009) 29: 447 456 DOI 10.1007/s10801-008-0145-0 A modular absolute bound condition for primitive association schemes Akihide Hanaki Ilia Ponomarenko Received: 29 September 2007 / Accepted:
More informationСИБИРСКИЕ ЭЛЕКТРОННЫЕ МАТЕМАТИЧЕСКИЕ ИЗВЕСТИЯ Siberian Electronic Mathematical Reports
S e R ISSN 1813-3304 СИБИРСКИЕ ЭЛЕКТРОННЫЕ МАТЕМАТИЧЕСКИЕ ИЗВЕСТИЯ Siberian Electronic athematical Reports http://semr.math.nsc.ru Том 13 стр. 1271 1282 2016 УДК 512.542.7 DOI 10.17377/semi.2016.13.099
More informationarxiv: v2 [math.gr] 7 Nov 2015
ON SCHUR 3-GROUPS GRIGORY RYABOV arxiv:1502.04615v2 [math.gr] 7 Nov 2015 Abstract. Let G be a finite group. If Γ is a permutation group with G right Γ Sym(G) and S is the set of orbits of the stabilizer
More informationQ N id β. 2. Let I and J be ideals in a commutative ring A. Give a simple description of
Additional Problems 1. Let A be a commutative ring and let 0 M α N β P 0 be a short exact sequence of A-modules. Let Q be an A-module. i) Show that the naturally induced sequence is exact, but that 0 Hom(P,
More informationFinite Fields. [Parts from Chapter 16. Also applications of FTGT]
Finite Fields [Parts from Chapter 16. Also applications of FTGT] Lemma [Ch 16, 4.6] Assume F is a finite field. Then the multiplicative group F := F \ {0} is cyclic. Proof Recall from basic group theory
More informationThe Galois group of a polynomial f(x) K[x] is the Galois group of E over K where E is a splitting field for f(x) over K.
The third exam will be on Monday, April 9, 013. The syllabus for Exam III is sections 1 3 of Chapter 10. Some of the main examples and facts from this material are listed below. If F is an extension field
More informationGalois Theory, summary
Galois Theory, summary Chapter 11 11.1. UFD, definition. Any two elements have gcd 11.2 PID. Every PID is a UFD. There are UFD s which are not PID s (example F [x, y]). 11.3 ED. Every ED is a PID (and
More informationAlgebra. Travis Dirle. December 4, 2016
Abstract Algebra 2 Algebra Travis Dirle December 4, 2016 2 Contents 1 Groups 1 1.1 Semigroups, Monoids and Groups................ 1 1.2 Homomorphisms and Subgroups................. 2 1.3 Cyclic Groups...........................
More informationFINITE GROUP THEORY: SOLUTIONS FALL MORNING 5. Stab G (l) =.
FINITE GROUP THEORY: SOLUTIONS TONY FENG These are hints/solutions/commentary on the problems. They are not a model for what to actually write on the quals. 1. 2010 FALL MORNING 5 (i) Note that G acts
More informationGraduate Preliminary Examination
Graduate Preliminary Examination Algebra II 18.2.2005: 3 hours Problem 1. Prove or give a counter-example to the following statement: If M/L and L/K are algebraic extensions of fields, then M/K is algebraic.
More informationSupplement. Dr. Bob s Modern Algebra Glossary Based on Fraleigh s A First Course on Abstract Algebra, 7th Edition, Sections 0 through IV.
Glossary 1 Supplement. Dr. Bob s Modern Algebra Glossary Based on Fraleigh s A First Course on Abstract Algebra, 7th Edition, Sections 0 through IV.23 Abelian Group. A group G, (or just G for short) is
More informationAlgebra-I, Fall Solutions to Midterm #1
Algebra-I, Fall 2018. Solutions to Midterm #1 1. Let G be a group, H, K subgroups of G and a, b G. (a) (6 pts) Suppose that ah = bk. Prove that H = K. Solution: (a) Multiplying both sides by b 1 on the
More informationLecture 7.3: Ring homomorphisms
Lecture 7.3: Ring homomorphisms Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 4120, Modern Algebra M. Macauley (Clemson) Lecture 7.3:
More informationProjective and Injective Modules
Projective and Injective Modules Push-outs and Pull-backs. Proposition. Let P be an R-module. The following conditions are equivalent: (1) P is projective. (2) Hom R (P, ) is an exact functor. (3) Every
More informationFIELD THEORY. Contents
FIELD THEORY MATH 552 Contents 1. Algebraic Extensions 1 1.1. Finite and Algebraic Extensions 1 1.2. Algebraic Closure 5 1.3. Splitting Fields 7 1.4. Separable Extensions 8 1.5. Inseparable Extensions
More informationNOTES FOR DRAGOS: MATH 210 CLASS 12, THURS. FEB. 22
NOTES FOR DRAGOS: MATH 210 CLASS 12, THURS. FEB. 22 RAVI VAKIL Hi Dragos The class is in 381-T, 1:15 2:30. This is the very end of Galois theory; you ll also start commutative ring theory. Tell them: midterm
More informationABELIAN HOPF GALOIS STRUCTURES ON PRIME-POWER GALOIS FIELD EXTENSIONS
ABELIAN HOPF GALOIS STRUCTURES ON PRIME-POWER GALOIS FIELD EXTENSIONS S. C. FEATHERSTONHAUGH, A. CARANTI, AND L. N. CHILDS Abstract. The main theorem of this paper is that if (N, +) is a finite abelian
More informationSelected exercises from Abstract Algebra by Dummit and Foote (3rd edition).
Selected exercises from Abstract Algebra by Dummit and Foote (3rd edition). Bryan Félix Abril 12, 2017 Section 14.2 Exercise 3. Determine the Galois group of (x 2 2)(x 2 3)(x 2 5). Determine all the subfields
More informationExercises MAT2200 spring 2013 Ark 4 Homomorphisms and factor groups
Exercises MAT2200 spring 2013 Ark 4 Homomorphisms and factor groups This Ark concerns the weeks No. (Mar ) and No. (Mar ). Plans until Eastern vacations: In the book the group theory included in the curriculum
More informationAbstract Algebra II Groups ( )
Abstract Algebra II Groups ( ) Melchior Grützmann / melchiorgfreehostingcom/algebra October 15, 2012 Outline Group homomorphisms Free groups, free products, and presentations Free products ( ) Definition
More informationPermutation representations and rational irreducibility
Permutation representations and rational irreducibility John D. Dixon School of Mathematics and Statistics Carleton University, Ottawa, Canada March 30, 2005 Abstract The natural character π of a finite
More informationProblem 1. Let I and J be ideals in a ring commutative ring R with 1 R. Recall
I. Take-Home Portion: Math 350 Final Exam Due by 5:00pm on Tues. 5/12/15 No resources/devices other than our class textbook and class notes/handouts may be used. You must work alone. Choose any 5 problems
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 informationSchool of Mathematics and Statistics. MT5836 Galois Theory. Handout 0: Course Information
MRQ 2017 School of Mathematics and Statistics MT5836 Galois Theory Handout 0: Course Information Lecturer: Martyn Quick, Room 326. Prerequisite: MT3505 (or MT4517) Rings & Fields Lectures: Tutorials: Mon
More informationEssential idempotents in group algebras and in Coding Theory
in group algebras and in Coding Theory César Polcino Milies Universidade de São Paulo and Universidade Federal do ABC NON COMMUTATIVE RINGS AND APPLICATIONS IV, LENS, 2015 Definition A linear code C F
More informationHonors Algebra 4, MATH 371 Winter 2010 Assignment 4 Due Wednesday, February 17 at 08:35
Honors Algebra 4, MATH 371 Winter 2010 Assignment 4 Due Wednesday, February 17 at 08:35 1. Let R be a commutative ring with 1 0. (a) Prove that the nilradical of R is equal to the intersection of the prime
More informationVector Bundles vs. Jesko Hüttenhain. Spring Abstract
Vector Bundles vs. Locally Free Sheaves Jesko Hüttenhain Spring 2013 Abstract Algebraic geometers usually switch effortlessly between the notion of a vector bundle and a locally free sheaf. I will define
More informationRings and Fields Theorems
Rings and Fields Theorems Rajesh Kumar PMATH 334 Intro to Rings and Fields Fall 2009 October 25, 2009 12 Rings and Fields 12.1 Definition Groups and Abelian Groups Let R be a non-empty set. Let + and (multiplication)
More informationList of topics for the preliminary exam in algebra
List of topics for the preliminary exam in algebra 1 Basic concepts 1. Binary relations. Reflexive, symmetric/antisymmetryc, and transitive relations. Order and equivalence relations. Equivalence classes.
More informationALGEBRA QUALIFYING EXAM SPRING 2012
ALGEBRA QUALIFYING EXAM SPRING 2012 Work all of the problems. Justify the statements in your solutions by reference to specific results, as appropriate. Partial credit is awarded for partial solutions.
More information9. Finite fields. 1. Uniqueness
9. Finite fields 9.1 Uniqueness 9.2 Frobenius automorphisms 9.3 Counting irreducibles 1. Uniqueness Among other things, the following result justifies speaking of the field with p n elements (for prime
More informationMath 120: Homework 6 Solutions
Math 120: Homewor 6 Solutions November 18, 2018 Problem 4.4 # 2. Prove that if G is an abelian group of order pq, where p and q are distinct primes then G is cyclic. Solution. By Cauchy s theorem, G has
More informationSymmetries of a q-ary Hamming Code
Symmetries of a q-ary Hamming Code Evgeny V. Gorkunov Novosibirsk State University Algebraic and Combinatorial Coding Theory Akademgorodok, Novosibirsk, Russia September 5 11, 2010
More informationLecture 8: The Field B dr
Lecture 8: The Field B dr October 29, 2018 Throughout this lecture, we fix a perfectoid field C of characteristic p, with valuation ring O C. Fix an element π C with 0 < π C < 1, and let B denote the completion
More informationMath 201C Homework. Edward Burkard. g 1 (u) v + f 2(u) g 2 (u) v2 + + f n(u) a 2,k u k v a 1,k u k v + k=0. k=0 d
Math 201C Homework Edward Burkard 5.1. Field Extensions. 5. Fields and Galois Theory Exercise 5.1.7. If v is algebraic over K(u) for some u F and v is transcendental over K, then u is algebraic over K(v).
More informationAlgebra Ph.D. Entrance Exam Fall 2009 September 3, 2009
Algebra Ph.D. Entrance Exam Fall 2009 September 3, 2009 Directions: Solve 10 of the following problems. Mark which of the problems are to be graded. Without clear indication which problems are to be graded
More informationFROM GROUPS TO GALOIS Amin Witno
WON Series in Discrete Mathematics and Modern Algebra Volume 6 FROM GROUPS TO GALOIS Amin Witno These notes 1 have been prepared for the students at Philadelphia University (Jordan) who are taking the
More informationMATHEMATICS COMPREHENSIVE EXAM: IN-CLASS COMPONENT
MATHEMATICS COMPREHENSIVE EXAM: IN-CLASS COMPONENT The following is the list of questions for the oral exam. At the same time, these questions represent all topics for the written exam. The procedure for
More informationSylow subgroups of GL(3,q)
Jack Schmidt We describe the Sylow p-subgroups of GL(n, q) for n 4. These were described in (Carter & Fong, 1964) and (Weir, 1955). 1 Overview The groups GL(n, q) have three types of Sylow p-subgroups:
More informationA PROOF OF BURNSIDE S p a q b THEOREM
A PROOF OF BURNSIDE S p a q b THEOREM OBOB Abstract. We prove that if p and q are prime, then any group of order p a q b is solvable. Throughout this note, denote by A the set of algebraic numbers. We
More informationSUMMARY ALGEBRA I LOUIS-PHILIPPE THIBAULT
SUMMARY ALGEBRA I LOUIS-PHILIPPE THIBAULT Contents 1. Group Theory 1 1.1. Basic Notions 1 1.2. Isomorphism Theorems 2 1.3. Jordan- Holder Theorem 2 1.4. Symmetric Group 3 1.5. Group action on Sets 3 1.6.
More informationφ(xy) = (xy) n = x n y n = φ(x)φ(y)
Groups 1. (Algebra Comp S03) Let A, B and C be normal subgroups of a group G with A B. If A C = B C and AC = BC then prove that A = B. Let b B. Since b = b1 BC = AC, there are a A and c C such that b =
More informationBASIC GROUP THEORY : G G G,
BASIC GROUP THEORY 18.904 1. Definitions Definition 1.1. A group (G, ) is a set G with a binary operation : G G G, and a unit e G, possessing the following properties. (1) Unital: for g G, we have g e
More informationSome algebraic number theory and the reciprocity map
Some algebraic number theory and the reciprocity map Ervin Thiagalingam September 28, 2015 Motivation In Weinstein s paper, the main problem is to find a rule (reciprocity law) for when an irreducible
More informationLarge Automorphism Groups of Algebraic Curves in Positive Characteristic. BMS-LMS Conference
Large Automorphism Groups of Algebraic Curves in Positive Characteristic Massimo Giulietti (Università degli Studi di Perugia) BMS-LMS Conference December 4-5, 2009 Leuven Notation and Terminology K algebraically
More informationWinter School on Galois Theory Luxembourg, February INTRODUCTION TO PROFINITE GROUPS Luis Ribes Carleton University, Ottawa, Canada
Winter School on alois Theory Luxembourg, 15-24 February 2012 INTRODUCTION TO PROFINITE ROUPS Luis Ribes Carleton University, Ottawa, Canada LECTURE 2 2.1 ENERATORS OF A PROFINITE ROUP 2.2 FREE PRO-C ROUPS
More informationAlgebraic Structures Exam File Fall 2013 Exam #1
Algebraic Structures Exam File Fall 2013 Exam #1 1.) Find all four solutions to the equation x 4 + 16 = 0. Give your answers as complex numbers in standard form, a + bi. 2.) Do the following. a.) Write
More informationThe number of different reduced complete sets of MOLS corresponding to the Desarguesian projective planes
The number of different reduced complete sets of MOLS corresponding to the Desarguesian projective planes Vrije Universiteit Brussel jvpoucke@vub.ac.be joint work with K. Hicks, G.L. Mullen and L. Storme
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 information12. Hilbert Polynomials and Bézout s Theorem
12. Hilbert Polynomials and Bézout s Theorem 95 12. Hilbert Polynomials and Bézout s Theorem After our study of smooth cubic surfaces in the last chapter, let us now come back to the general theory of
More informationGroup Theory. 1. Show that Φ maps a conjugacy class of G into a conjugacy class of G.
Group Theory Jan 2012 #6 Prove that if G is a nonabelian group, then G/Z(G) is not cyclic. Aug 2011 #9 (Jan 2010 #5) Prove that any group of order p 2 is an abelian group. Jan 2012 #7 G is nonabelian nite
More informationChapter 3. Rings. The basic commutative rings in mathematics are the integers Z, the. Examples
Chapter 3 Rings Rings are additive abelian groups with a second operation called multiplication. The connection between the two operations is provided by the distributive law. Assuming the results of Chapter
More information1 Finite abelian groups
Last revised: May 16, 2014 A.Miller M542 www.math.wisc.edu/ miller/ Each Problem is due one week from the date it is assigned. Do not hand them in early. Please put them on the desk in front of the room
More informationALGEBRA EXERCISES, PhD EXAMINATION LEVEL
ALGEBRA EXERCISES, PhD EXAMINATION LEVEL 1. Suppose that G is a finite group. (a) Prove that if G is nilpotent, and H is any proper subgroup, then H is a proper subgroup of its normalizer. (b) Use (a)
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 informationTheorem 5.3. Let E/F, E = F (u), be a simple field extension. Then u is algebraic if and only if E/F is finite. In this case, [E : F ] = deg f u.
5. Fields 5.1. Field extensions. Let F E be a subfield of the field E. We also describe this situation by saying that E is an extension field of F, and we write E/F to express this fact. If E/F is a field
More information1 Notations and Statement of the Main Results
An introduction to algebraic fundamental groups 1 Notations and Statement of the Main Results Throughout the talk, all schemes are locally Noetherian. All maps are of locally finite type. There two main
More information(1) A frac = b : a, b A, b 0. We can define addition and multiplication of fractions as we normally would. a b + c d
The Algebraic Method 0.1. Integral Domains. Emmy Noether and others quickly realized that the classical algebraic number theory of Dedekind could be abstracted completely. In particular, rings of integers
More informationB Sc MATHEMATICS ABSTRACT ALGEBRA
UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION B Sc MATHEMATICS (0 Admission Onwards) V Semester Core Course ABSTRACT ALGEBRA QUESTION BANK () Which of the following defines a binary operation on Z
More informationCover Page. The handle holds various files of this Leiden University dissertation.
Cover Page The handle http://hdl.handle.net/1887/20310 holds various files of this Leiden University dissertation. Author: Jansen, Bas Title: Mersenne primes and class field theory Date: 2012-12-18 Chapter
More informationMath 418 Algebraic Geometry Notes
Math 418 Algebraic Geometry Notes 1 Affine Schemes Let R be a commutative ring with 1. Definition 1.1. The prime spectrum of R, denoted Spec(R), is the set of prime ideals of the ring R. Spec(R) = {P R
More informationALGEBRA PH.D. QUALIFYING EXAM September 27, 2008
ALGEBRA PH.D. QUALIFYING EXAM September 27, 2008 A passing paper consists of four problems solved completely plus significant progress on two other problems; moreover, the set of problems solved completely
More informationALGEBRA 11: Galois theory
Galois extensions Exercise 11.1 (!). Consider a polynomial P (t) K[t] of degree n with coefficients in a field K that has n distinct roots in K. Prove that the ring K[t]/P of residues modulo P is isomorphic
More informationABSTRACT ALGEBRA WITH APPLICATIONS
ABSTRACT ALGEBRA WITH APPLICATIONS IN TWO VOLUMES VOLUME I VECTOR SPACES AND GROUPS KARLHEINZ SPINDLER Darmstadt, Germany Marcel Dekker, Inc. New York Basel Hong Kong Contents f Volume I Preface v VECTOR
More information5 Group theory. 5.1 Binary operations
5 Group theory This section is an introduction to abstract algebra. This is a very useful and important subject for those of you who will continue to study pure mathematics. 5.1 Binary operations 5.1.1
More informationRiemann surfaces with extra automorphisms and endomorphism rings of their Jacobians
Riemann surfaces with extra automorphisms and endomorphism rings of their Jacobians T. Shaska Oakland University Rochester, MI, 48309 April 14, 2018 Problem Let X be an algebraic curve defined over a field
More informationFoundational Aspects of Linear Codes: 3. Extension property: sufficient conditions
Foundational Aspects of Linear Codes: 3. Extension property: sufficient conditions Jay A. Wood Department of Mathematics Western Michigan University http://homepages.wmich.edu/ jwood/ On the Algebraic
More informationInverse Galois Problem for C(t)
Inverse Galois Problem for C(t) Padmavathi Srinivasan PuMaGraSS March 2, 2012 Outline 1 The problem 2 Compact Riemann Surfaces 3 Covering Spaces 4 Connection to field theory 5 Proof of the Main theorem
More informationCriterion of maximal period of a trinomial over nontrivial Galois ring of odd characteristic
Criterion of maximal period of a trinomial over nontrivial Galois ring of odd characteristic V.N.Tsypyschev and Ju.S.Vinogradova Russian State Social University, 4,W.Pik Str., Moscow, Russia Abstract In
More informationk-cleaning of Dessin d Enfants
k-cleaning of Dessin d Enfants Gabrielle Melamed, Jonathan Pham, Austin Wei Willamette University Mathematics Consortium REU August 4, 2017 Outline Motivation Belyi Maps Introduction and Definitions Dessins
More informationBasic Definitions: Group, subgroup, order of a group, order of an element, Abelian, center, centralizer, identity, inverse, closed.
Math 546 Review Exam 2 NOTE: An (*) at the end of a line indicates that you will not be asked for the proof of that specific item on the exam But you should still understand the idea and be able to apply
More informationGALOIS THEORY BRIAN OSSERMAN
GALOIS THEORY BRIAN OSSERMAN Galois theory relates the theory of field extensions to the theory of groups. It provides a powerful tool for studying field extensions, and consequently, solutions to polynomial
More informationGalois theory. Philippe H. Charmoy supervised by Prof Donna M. Testerman
Galois theory Philippe H. Charmoy supervised by Prof Donna M. Testerman Autumn semester 2008 Contents 0 Preliminaries 4 0.1 Soluble groups........................... 4 0.2 Field extensions...........................
More informationDynamics of finite linear cellular automata over Z N
Dynamics of finite linear cellular automata over Z N F. Mendivil, D. Patterson September 9, 2009 Abstract We investigate the behaviour of linear cellular automata with state space Z N and only finitely
More informationALGEBRA QUALIFYING EXAM PROBLEMS LINEAR ALGEBRA
ALGEBRA QUALIFYING EXAM PROBLEMS LINEAR ALGEBRA Kent State University Department of Mathematical Sciences Compiled and Maintained by Donald L. White Version: August 29, 2017 CONTENTS LINEAR ALGEBRA AND
More informationPermutation groups/1. 1 Automorphism groups, permutation groups, abstract
Permutation groups Whatever you have to do with a structure-endowed entity Σ try to determine its group of automorphisms... You can expect to gain a deep insight into the constitution of Σ in this way.
More informationMath 429/581 (Advanced) Group Theory. Summary of Definitions, Examples, and Theorems by Stefan Gille
Math 429/581 (Advanced) Group Theory Summary of Definitions, Examples, and Theorems by Stefan Gille 1 2 0. Group Operations 0.1. Definition. Let G be a group and X a set. A (left) operation of G on X is
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 informationLecture 6: Etale Fundamental Group
Lecture 6: Etale Fundamental Group October 5, 2014 1 Review of the topological fundamental group and covering spaces 1.1 Topological fundamental group Suppose X is a path-connected topological space, and
More informationGroup Algebras with the Bounded Splitting Property. Derek J.S. Robinson. Ischia Group Theory 2014
Group Algebras with the Bounded Splitting Property Derek J.S. Robinson University of Illinois at Urbana-Champaign Ischia Group Theory 2014 (Joint work with Blas Torrecillas, University of Almería) Derek
More informationSelected exercises from Abstract Algebra by Dummit and Foote (3rd edition).
Selected exercises from Abstract Algebra by Dummit Foote (3rd edition). Bryan Félix Abril 12, 2017 Section 4.1 Exercise 1. Let G act on the set A. Prove that if a, b A b = ga for some g G, then G b = gg
More information12. Projective modules The blanket assumptions about the base ring k, the k-algebra A, and A-modules enumerated at the start of 11 continue to hold.
12. Projective modules The blanket assumptions about the base ring k, the k-algebra A, and A-modules enumerated at the start of 11 continue to hold. 12.1. Indecomposability of M and the localness of End
More information