Louis Halle Rowen is a professor emeritus in the Department of Mathematics, Bar-Ilan University. He received his PhD from Yale University. His research specialty is noncommutative algebra, in particular division algebras as well as the structure of rings. He is an enthusiastic cellist, having soloed with the Jerusalem Symphony. Prof. Rowen is a fellow of the American Mathematics Society, and has been awarded the Landau Prize, Van Buren Mathematics Prize, and Van Amringe Mathematics Prize.Uzi Vishne is a professor in the Department of Mathematics, Bar-Ilan University. He holds a PhD from Bar-Ilan University. He is managing editor of the Israel Mathematics Conference Proceedings (IMCP) book series. He has authored or co-authored over seventy papers in algebra, arithmetic, combinatorics, and their applications.

• 1 Monoids and Groups 1.1 Examples of Groups and MonoidsWhen Is a Monoid a Group?1.2 Exercises2 Lagrange’s Theorem, Cosets, and an Application to Number Theory 2.1 Cosets2.2 Fermat’s Little Theorem2.3 Exercises3 Cauchy’s Theorem: Showing that a Number Is Greater Than 13.1 The Exponent3.2 The symmetric group Sn: Our Main Example3.3 The Product of Two Subgroups3.4 Exercises4 Structure of Groups: Homomorphisms, Isomorphisms, and Invariants 4.1 Homomorphic Images4.2 Exercises5 Normal Subgroups: The Building Blocks of the Structure Theory 5.1 The Residue Group5.2 Noether’s Isomorphism Theorems5.3 Conjugates in Sn5.4 The Alternating Group5.5 Exercises6 Classifying Groups: Cyclic Groups and Direct Products 6.1 Cyclic Groups6.2 Generators of a Group6.3 Direct Products6.4 Application: Some Algebraic Cryptosystems6.5 Exercises7 Finite Abelian Groups 7.1 Abelian p-Groups7.2 Proof of the Fundamental Theorem for Finite abelian Groups 7.3 The Classification of Finite abelian Groups7.4 Exercises8 Generators and Relations 8.1 Description of Groups of Low Order8.3 Exercises9 When Is a Group a Group? (Cayley’s Theorem) 9.1 The Generalized Cayley Theorem9.2 Introduction to Group Representations9.3 Exercises10 Conjugacy Classes and the Class Equation 10.1 The Center of a Group10.2 Exercises11 Sylow Subgroups 11.1 Groups of Order Less Than 6011.2 Finite Simple Groups11.3 Exercises12 Solvable Groups: What Could Be Simpler? 12.1 Commutators12.2 Solvable Groups12.3 Automorphisms of Groups12.4 Exercises13 Groups of Matrices 13.1 Exercises14 An Introduction to Rings 14.1 Domains and Skew Fields14.2 Left Ideals14.3 Exercises15 The Structure Theory of Rings 15.1 Ideals15.2 Noether’s Isomorphism Theorems for Rings15.3 Exercises16 The Field of Fractions: A Study in Generalization 16.1 Intermediate Rings16.2 Exercises17 Polynomials and Euclidean Domains 17.1 The Ring of Polynomials17.2 Euclidean Domains17.3 Unique Factorization17.4 Exercises18 Principal Ideal Domains: Induction without Numbers 18.1 Prime Ideals18.2 Noetherian RingsExercises19 Roots of Polynomials 19.1 Finite Subgroups of Fields19.2 Primitive Roots of 119.3 Exercises20 Applications: Famous Results from Number Theory 20.1 A Theorem of Fermat20.2 Addendum: “Fermat’s Last Theorem”20.3 Exercises21 Irreducible Polynomials 21.1 Polynomials over UFDs21.2 Eisenstein’s Criterion21.3 Exercises22 Field Extensions: Creating Roots of Polynomials 22.1 Algebraic Elements22.2 Finite Field Extensions22.3 Exercises23 The Geometric Problems of Antiquity 23.1 Construction by Straight Edge and Compass23.2 Algebraic Description of Constructibility23.3 Solution of the Geometric Problems of Antiquity23.4 Exercises24 Adjoining Roots to Polynomials: Splitting Fields 24.1 Splitting Fields24.2 Separable Polynomials and Separable Extensions24.3 Exercises25 Finite Fields 25.1 Uniqueness25.2 Existence25.3 Exercises26 The Galois Correspondence 26.1 The Galois Group of a Field Extension26.2 The Galois Group and Intermediate Fields26.3 Exercises27 Applications of the Galois Correspondence 27.1 Finite Separable Field Extensions and the Normal Closure27.2 The Galois Group of a Polynomial27.3 Constructible n-gons27.4 Finite Fields27.5 The Fundamental Theorem of Algebra27.6 Exercises28 Solving Equations by Radicals 28.1 Radical Extensions28.2 Solvable Galois Groups28.3 Computing the Galois Group28.4 Exercises29 Integral Extensions 29.1 Exercises30 Group Representations and their Characters 30.1 Exercises31 Transcendental Numbers: e and π 31.1 Transcendence of e31.2 Transcendence of π32 Skew Field Theory 32.1 The Quaternion Algebra32.2 Polynomials over Skew Fields32.3 Structure Theorems for Skew Fields32.4 Exercises33 Where Do We Go From Here?33.1 Modules33.2 Matrix Algebras and their Substructures33.3 Nonassociative Rings and Algebras33.4 Hyperfields33.5 Exercises

Louis Halle Rowen is a professor emeritus in the Department of Mathematics, Bar-Ilan University. He received his PhD from Yale University. His research specialty is noncommutative algebra, in particular division algebras as well as the structure of rings. He is an enthusiastic cellist, having soloed with the Jerusalem Symphony. Prof. Rowen is a fellow of the American Mathematics Society, and has been awarded the Landau Prize, Van Buren Mathematics Prize, and Van Amringe Mathematics Prize.Uzi Vishne is a professor in the Department of Mathematics, Bar-Ilan University. He holds a PhD from Bar-Ilan University. He is managing editor of the Israel Mathematics Conference Proceedings (IMCP) book series. He has authored or co-authored over seventy papers in algebra, arithmetic, combinatorics, and their applications.

• 1 Monoids and Groups 1.1 Examples of Groups and MonoidsWhen Is a Monoid a Group?1.2 Exercises2 Lagrange’s Theorem, Cosets, and an Application to Number Theory 2.1 Cosets2.2 Fermat’s Little Theorem2.3 Exercises3 Cauchy’s Theorem: Showing that a Number Is Greater Than 13.1 The Exponent3.2 The symmetric group Sn: Our Main Example3.3 The Product of Two Subgroups3.4 Exercises4 Structure of Groups: Homomorphisms, Isomorphisms, and Invariants 4.1 Homomorphic Images4.2 Exercises5 Normal Subgroups: The Building Blocks of the Structure Theory 5.1 The Residue Group5.2 Noether’s Isomorphism Theorems5.3 Conjugates in Sn5.4 The Alternating Group5.5 Exercises6 Classifying Groups: Cyclic Groups and Direct Products 6.1 Cyclic Groups6.2 Generators of a Group6.3 Direct Products6.4 Application: Some Algebraic Cryptosystems6.5 Exercises7 Finite Abelian Groups 7.1 Abelian p-Groups7.2 Proof of the Fundamental Theorem for Finite abelian Groups 7.3 The Classification of Finite abelian Groups7.4 Exercises8 Generators and Relations 8.1 Description of Groups of Low Order8.3 Exercises9 When Is a Group a Group? (Cayley’s Theorem) 9.1 The Generalized Cayley Theorem9.2 Introduction to Group Representations9.3 Exercises10 Conjugacy Classes and the Class Equation 10.1 The Center of a Group10.2 Exercises11 Sylow Subgroups 11.1 Groups of Order Less Than 6011.2 Finite Simple Groups11.3 Exercises12 Solvable Groups: What Could Be Simpler? 12.1 Commutators12.2 Solvable Groups12.3 Automorphisms of Groups12.4 Exercises13 Groups of Matrices 13.1 Exercises14 An Introduction to Rings 14.1 Domains and Skew Fields14.2 Left Ideals14.3 Exercises15 The Structure Theory of Rings 15.1 Ideals15.2 Noether’s Isomorphism Theorems for Rings15.3 Exercises16 The Field of Fractions: A Study in Generalization 16.1 Intermediate Rings16.2 Exercises17 Polynomials and Euclidean Domains 17.1 The Ring of Polynomials17.2 Euclidean Domains17.3 Unique Factorization17.4 Exercises18 Principal Ideal Domains: Induction without Numbers 18.1 Prime Ideals18.2 Noetherian RingsExercises19 Roots of Polynomials 19.1 Finite Subgroups of Fields19.2 Primitive Roots of 119.3 Exercises20 Applications: Famous Results from Number Theory 20.1 A Theorem of Fermat20.2 Addendum: “Fermat’s Last Theorem”20.3 Exercises21 Irreducible Polynomials 21.1 Polynomials over UFDs21.2 Eisenstein’s Criterion21.3 Exercises22 Field Extensions: Creating Roots of Polynomials 22.1 Algebraic Elements22.2 Finite Field Extensions22.3 Exercises23 The Geometric Problems of Antiquity 23.1 Construction by Straight Edge and Compass23.2 Algebraic Description of Constructibility23.3 Solution of the Geometric Problems of Antiquity23.4 Exercises24 Adjoining Roots to Polynomials: Splitting Fields 24.1 Splitting Fields24.2 Separable Polynomials and Separable Extensions24.3 Exercises25 Finite Fields 25.1 Uniqueness25.2 Existence25.3 Exercises26 The Galois Correspondence 26.1 The Galois Group of a Field Extension26.2 The Galois Group and Intermediate Fields26.3 Exercises27 Applications of the Galois Correspondence 27.1 Finite Separable Field Extensions and the Normal Closure27.2 The Galois Group of a Polynomial27.3 Constructible n-gons27.4 Finite Fields27.5 The Fundamental Theorem of Algebra27.6 Exercises28 Solving Equations by Radicals 28.1 Radical Extensions28.2 Solvable Galois Groups28.3 Computing the Galois Group28.4 Exercises29 Integral Extensions 29.1 Exercises30 Group Representations and their Characters 30.1 Exercises31 Transcendental Numbers: e and π 31.1 Transcendence of e31.2 Transcendence of π32 Skew Field Theory 32.1 The Quaternion Algebra32.2 Polynomials over Skew Fields32.3 Structure Theorems for Skew Fields32.4 Exercises33 Where Do We Go From Here?33.1 Modules33.2 Matrix Algebras and their Substructures33.3 Nonassociative Rings and Algebras33.4 Hyperfields33.5 Exercises