Instructors Info

Lecture: Maria Angelica Cueto
Email: cueto.5@osu.edu

Office Hours

Thursday 2:00-4:00pm
room to be determined.

Time and Location

Lecture: M-F 11:30am-12:25pm
Baker Systems Engineering (BE) 130.

Overview and course content

 This is part one of a very intense year-long course on Abstract Agebra. There will be lectures five days a week, with occasional days for problem solving session, if needed

The course will be divided in three parts. In the first part, we will study Groups and their algebraic structures. In the second part, we will focus on Ring Theory, with emphasis on the commutative setting. Finally, in the last part, we will discuss Polynomial Rings as our main example, as well as various applications to Elementary Number Theory.

Content:
  • Group Theory: Definitions, examples, basic properties. Subgroups, cosets; Lagrange's theorem. Normal subgroups, quotient groups. Simple groups. Subnormal,normal, and composition series. Conjugateelements, conjugacy classes. Normalizers and centralizers. Group homomorphisms. The isomorphism theorems. Actions of groups, orbits and stabilizers; actions of groups on themselves. Direct products of groups. Classification of finitely generated abelian groups. Groups of automorphisms. Semidirect products, permutation group, simple groups, simplicity of alternating groups. Sylow Theorems. Groups of orders p, p2, p3, pq, p2q. Commutator calculus, composition series, Jordan-Hölder theorem, nilpotent and solvable groups.

  • Ring Theory: Definitions, examples, basic properties. Rings of fractions. Ring homomorphisms, ideals, quotient rings. Isomorphism theorems. Prime and maximal ideals. Nilradical, primary ideals. Direct products of rings, the Chinese remainder theorem. Prime and irreducible elements. Noetherian and Artinian rings, Euclidean, principal ideal, unique factorization domains. Quadratic integer rings.

  • Polynomial Rings: Roots of polynomials. Polynomials over UFDs and Gauss’s lemma. Irreducibility criteria. Symmetric polynomials. Polynomials in several variables, Gröbner bases (if time allows).

Prerequisites: Grade C or better in Honors Linear Algebra and Differential Equations (Math 5520H), or permission of the Deparment.

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Syllabus and references

 Textbook: D.S. Dummit and R.M. Foote, \emph{Abstract Algebra}, 3rd edition, Wiley. We will cover Chapters 1-9 and 15 (groups,rings, polynomials) this semester.

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Exams
  • Midterm 1: Tuesday September 17, 2024. Topics: Group Theory I (Sections 1.1-1.7, 2.1-2.5, 3.1-3.3, 4.1-4.3, 6.2)
  • Midterm 2: Friday October 18, 2024. Topics: Group Theory II (Sections 3.4-3.5, 4.4-4.6, 5.1-5.5, 6.1-6.2)
  • Midterm 3: Tuesday November 19, 2024. Topics: Ring Theory (Sections 7.1-7.6, 8.1-8.3, 9.1-9.4, 15.1, 15.4, 16.1)
  • Final exam (CUMULATIVE): Thursday December 12, 2024 (10:00am-11:45am, location TBD) In addition to the previous topics, the final exam will include material from Sections 9.5 and 9.6.

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Homework

 There will be 11 homework problem sets. These will be posted as the semester progresses. The lowest homework score will be dropped.
Homeworks will be submitted in person, in class. No late homework will be accepted without medical excuse.

Participants can work in teams, but individual solutions must be submitted for grading and credit.
  • Homework 1 (due Friday August 30)
    • Problems for submission: 1.1: 25, 1.2: 4, 1.3: 16, 2.1:8, 2.4: 14.
    • Problems for practice and discussion:
      1.1: 6 (c,d,f), 7, 8, 14, 25, 31.
      1.2: 1, 3, 7, 18.
      1.3: 1, 5, 11, 15, 16, 17.
      2.1: 2 (a,b,e), 3, 5, 8, 12, 14.
      2.3: 21, 22, 23.
      2.4: 67, 7, 8, 13, 14.
    • Suggested Reading: Preliminary Section 0.3, especially Proposition 4, and Problems 10-14 (pages 11-12).

  • Homework 2 (due Friday September 6)
    • Problems for submission: 1.6: 9, 25, 3.1: 36, 41, 3.2: 18.
    • Problems for practice and discussion:
      1.6: 2, 4, 6-9, 17, 19, 20, 23, 25.
      3.1: 1, 3, 9, 11, 14, 17, 18, 25, 36, 40-42.
      3.2: 6, 8, 10, 14, 18, 19.
    • Suggested Reading: Section 1.5: definition of the quaternion group.
    • Bonus problem: Show that the quaternion group Q8 defined in Section 1.5 is non-abelian but all its subgroups are normal.

  • Homework 3 (due Friday September 13)
    • Problems for submission: 4.1: 2, 4, 10 (a, d), 4.3: 24, 26.
    • Problems for practice and discussion:
      1.7: 8-12, 2.2: 4-6, 9, 10.
      2.2: 2, 4-7, 9, 10.
      4.1: 1-6, 10.
      4.3: 6, 8, 9, 11, 17, 23-27, 33.
    • Suggested Reading: Section 3.5: Transpositions and the alternating group.

  • Homework 4 (due Friday September 27)
    • Problems for submission: 4.5: 14, 22, 32, 33, 6.2: 18.
    • Problems for practice and discussion:
      4.5: 4, 5, 8, 13-25 (try a few of these), 30, 32, 33, 36, 39, 40.
      6.2: 18-20, 22.
    • Bonus problem: Prove Theorem 3 of Lecture 21 (how to compute the structure constants fo a finite abelian p-group).

  • Homework 5 (due Friday October 4)
    • Problems for submission: 4.4: 12, 5.2: 1, 8, 5.5: 6, 22.
    • Problems for practice and discussion:
      4.4: 1, 3-5, 10, 12-14, 18, 19.
      5.1: 1, 4-6, 14.
      5.2: 1, 2, 7-9, 12, 13.
      5.5: 6-13, 15, 16, 22.
    • Suggested Reading: Examples on pages 181-184

  • Homework 6 (due Wednesday October 9)
    • Problems for submission: 3.4: 1, 2, 4, 6, 7
    • Problems for practice and discussion:
      3.4: 1, 2, 3, 4, 7, 6, 10, 12.
      6.1: 2, 3, 5, 6, 7, 12, 13.

  • Homework 7 (due Friday October 25)
    • Problems for submission: 7.1: 26, 27, 7.2: 3, 7, 7.3: 29, 33,
    • Problems for practice and discussion:
      7.1: 5-7, 11-16, 26, 27.
      7.2: 1-7.
      7.3: 10, 12-14, 21, 25, 26, 29, 33, 35.
    • Suggested Reading: Quadratic rings of integers (page 229).

  • Homework 8 (due Friday November 1)
    • Problems for submission: 7.4: 11, 15, 30, 7.5: 5, 7.6: 5.
    • Problems for practice and discussion:
      7.4: 4, 7-12, 14-19, 25, 26, 28, 30, 38, 39, 41.
      7.5: 2, 3-6.
      7.6: 1, 3-7.
    • Suggested Reading: Examples 8-11 of Section 7.6.

  • Homework 9 (due Friday November 8)
    • Problems for submission: 8.1: 4, 5, 8.2: 2, 4, 5.
    • Problems for practice and discussion:
      8.1: 3, 4, 5, 7, 8 (a for D=-7), 9, 11
      8.2: 2, 3, 4, 5, 8
    • Suggested Reading: Examples (3) and (4) from Section 8.1 (page 272), Example (2) on page 279 and Example on page 282.

  • Homework 10 (due Friday November 15)
    • Problems for submission:8.3: 8, 9.1: 6, 17, 9.2: 7, 8.
    • Problems for practice and discussion:
      8.3: 2, 3, 5, 6, 8
      9.1: 3-8, 10, 15-17
      9.2: 7, 8, 10.
    For Problem 17 in Section 9.1 you will need the following statements:
    Definition: A non-zero polynomial in n variables is homogeneous of degree k if the total degree (sum of exponents) of each monomial in it has degree k.
    Lemma: Every non-zero polynomial can be written (uniquely) as a sum of non-zero homogeneous polynomials of fixed degree (called its homogeneous components).

  • Homework 11 (due Monday December 2)
    • Problems for submission: 9.3: 3, 9.4: 13, 19(b,d), 9.6: 11, 19.
    • Problems for practice and discussion:
      9.3: 1, 3, 4.
      9.4: 1 (c), 2 (b,d), 7, 9, 12-14, 18, 19.
      9.6: 2, 3, 4, 5, 7, 9, 10-15, 16-19, 22-26.


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Lectures
  • Week 1:
    • Lecture 1 (Definition of a group, Examples: symmetric group, general linear group.), August 20, 2024.
    • Lecture 2 (Dihedral group, free group on two letters; Subgroups. Generating sets, Cyclic groups, Order of a group.), August 21, 2024.
    • Lecture 3 (Order of an element, Generators of the symmetric group, presentation of a group by generators and relations.), August 22, 2024.
    • Lecture 4 (Presentations of dihedral groups; Free groups; subgroups of finitely generated groups need not be finitely generated.), August 23, 2024.
  • Week 2:
    • Lecture 5 (Left cosets, index of a subgroup.), August 26, 2024.
    • Lecture 6 (Right cosets; Normal subgroups. Quotient groups.), August 27, 2024.
    • Lecture 7 (More examples of normal subgroups; groups given by generators and relations, Group homomorphisms. Isomorphisms, Kernel and images.), August 28, 2024.
    • Lecture 8 (Examples of group homomorphisms.), August 29, 2024.
    • Lecture 9 (Three Isomorphism Theorems, applications of the First Isomorphism Theorem.), August 30, 2024.
  • Week 3:
    • Lecture 10 (A quick review. Definition of group actions. Orbits, stabilizers and fixed points.), September 3, 2024.
    • Lecture 11 (More examples, Orbit-stabilizer correspondence.), September 4, 2024.
    • Lecture 12 (Some fun applications; Some counting results. Burnside's theorem on counting number of orbits.), September 5, 2024.
    • Lecture 13 (Free, faithful and transitive actions; Examples of group actions: Action via conjugation. Conjugacy classes, centralizers; alternative proof of Burnside's lemma.), September 6, 2024.
  • Week 4:
    • Lecture 14 (p-groups. Sylow theorems), September 9, 2024.
    • Lecture 15 (Proof of Sylow theorems, Examples: Sylow subgroups of some symmetric groups.), September 10, 2024.
    • Lecture 16 (More examples of Sylow subgroups; an application of Sylow theorems: proving a group is simple; p-groups are of order larger than p are never simple.), September 11, 2024.
    • Lecture 17 (More applications of Sylow Theorems: proving a group is not simple. Sylow subgroups of dihedral groups.), September 12, 2024.
    • Lecture 18 (Groups of size equal to the square of a prime number. Direct products. Abelian p-groups. Classification of finite abelian groups. Optional: examples of Sylow p-groups of linear groups over finite fields and symmetric groups.), September 13, 2024.
  • Week 5:
    • Lecture 19 (Direct product of groups; Classification of finite abelian groups: every finite abelian group is the direct product of its Sylow subgroups.), September 16, 2024.
    • Lecture 20 (Classification theorem for finite abelian p-groups with p a prime number: existence), September 17, 2024.
    • Lecture 21 (Classification theorem for finite abelian p-groups with p a prime number: uniqueness. How to determine the structure constants of an abelian p-group. Revisiting finite abelian groups - another classification theorem), September 20, 2024.
  • Week 6:
    • Lecture 22 (Generalized 3rd Isomorphim Theorem for groups. Semidirect products. Examples.), September 23, 2024.
    • Lecture 23 (Semidirect products via group actions.), September 24, 2024.
    • Lecture 24 (Equivalence between two constructions of semi-drect products. First examples of automorphism groups. Automorphic groups of cyclic groups are abelian.), September 25, 2024.
    • Lecture 25 (Automorphisms of cyclic groups. Reduction to the case when the group is a cyclic p-group. Groups of order p have cyclic automorphism groups), September 26, 2024.
    • Lecture 26 (Automorphisms of cyclic groups of order pr for r>1.), September 27, 2024.
  • Week 7:
    • Lecture 27 (Classification of grupos of order 18.), September 30, 2024.
    • Lecture 28 (Composition series. Basic properties of composition series. Jordan-Hölder series.), October 1, 2024.
    • Lecture 29 (General lemmas for Jordan-Hölder series. Schreier's Theorem. Uniqueness of Jordan-Hölder series up to equivalence. Zassenhaus lemma.), October 2, 2024.
    • Lecture 30 (Commutator series. Solvable groups - definitions and examples.), October 3, 2024.
    • Lecture 31 (Equivalent characterization of solvable groups. p-groups are solvable.), October 4, 2024.
  • Week 8:
    • Lecture 32 (Central series, nilpotent groups. Basic properties of nilpotent groups. p-groups are nilpotent), October 7, 2024.
    • Lecture 33 (Characterization of finite Nilpotent groups as direct products of p-groups. Normalizer of Sylow p-subgroups.), October 8, 2024.
    • Lecture 34 (Symmetric and alternating groups: generating sets, simplicity of An for n≥5, a presentation for Sn.), October 9 and 14, 2024.
  • Week 9:
    • Lecture 35 (Definition of rings. Examples. The endomorphism ring of an abelian group. Invertible elements. Zero divisors. Integral domains), October 15, 2024.
    • Lecture 36 (Ring homomorphisms. Kernel and Image of a ring homomorphism. Subrings. Ideals.), October 18, 2024.
  • Week 10:
    • Lecture 37 (More on ideals. Quotient rings. First isomorphism theorem for ring homomorphisms. Operations with left ideals: intersections, sum; left-ideals generated by subsets), October 21, 2024.
    • Lecture 38 (Operations with ideals: Product ideals; Coprime ideals; Direct product of rings), October 22, 2024.
    • Lecture 39 (Interaction between left/righ/2-sided ideals and ring homomorphisms, Second Isomorphism Theorem, All ideals of the polynomial ring K[X] with coefficients in a field and the ring of Gaussian integers Z[i] are principal.), October 23, 2024.
    • Lecture 40 (The characteristic of a ring, Principal rings. Prime and Maximal ideals. Examples: Prime and maximal ideals on Z and C[x].), October 24, 2024.
    • Lecture 41 (Geometry of commutative rings. Existence of maximal and prime ideals on commutative rings.), October 25, 2024.
  • Week 11:
    • Lecture 42 (Local rings: characterization of local rings; three main examples: quotients, completions and locatizations.), October 28, 2024.
    • Lecture 43 (Ring of fractions: definition and examples. Field of fractions.), October 29, 2024.
    • Lecture 44 (Localization pf a commutative ring at a prime ideal. Ideals in the ring of fractions.), October 30, 2024.
    • Lecture 45 (Prime ideals of S-1R, and their correspondence with prime ideals of R that are disjoint from S.), October 31, 2024.
    • Lecture 46 (Nilradical and Jacobson radical. Landscape of rings. Definition of Euclidean domains and first examples. Euclidean domains are PIDs. Quadratic field extensions of Q), November 1, 2024.
  • Week 12:
    • Lecture 47 (Ring of quadratic integers. Examples and characterization of which ones are Euclidean domains.), November 4, 2024.
    • Lecture 48 (Quadratic rings of integers for D = -1, -2, -3, -5, -7, -11.), November 6, 2024.
    • Lecture 49 (The Quadratic rings of integers for D = -19, is a PID but not a Euclidean domain. Dedekind-Hasse norms produce PIDs.), November 7, 2024.
    • Lecture 50 (Unique factorization domains, PIDs are UFDs, Euclidean domains are UFDs, examples, irreducible elements in UFDs or PIDs are prime), November 8, 2024.
  • Week 13:
    • Lecture 51 (Greatest Common divisors. Noetherian rings. Definitions and examples.), November 12, 2024.
    • Lecture 52 (Properties of Noetherian rings: quotients and ring of fractions of Noetherian rings are Noetherian. Hilbert Basis Theorem.), November 13, 2024.
    • Lecture 53 (Hilbert Basis Theorem (continued); Polynomials over UFDs, Gauss's lemma for irreducibility of primitive polynomials over UFDs.), November 14, 2024.
    • Lecture 54 (Gauss' lemma (continued). R UFD implies R[x] is also a UFD.), November 15, 2024.
  • Week 14:
    • Lecture 55 (Summary of UFDs, Noetherian domains and UFDs, subrings and quotient rings of UFDs need not be UFDs. Eisenstein's criterion for irreducibility of primitive polynomials over UFDs.), November 18, 2024.
    • Lecture 56 (Roots of polynomials and their multiplicities. Towards Groebner Bases I: why Groebner bases?; term orders of monomials, leading terms of multivariate polynomials), November 19, 2024.
    • Lecture 57 (Towards Groebner Bases II: examples of term orderes, well-ordering vs. 1 being the smallest monomial, definition of leading ideals and Gröbner bases; motivation for a multivariate division algorithm and an algorithm that determines if a set is a Gröbner bases without the need to precompute leading ideals.), November 22, 2024.
  • Week 15:
    • Lecture 58 (Towards Groebner Bases III: the multivariate division algorithm, existence of Gröbner bases, algorithm for membership test of polynomials in given ideals via Gröbner bases.), November 25, 2024.
    • Lecture 59 (Groebner bases IV: Buchberger's Theorem for detecting GRö.bner bases.), November 26, 2024.
  • Week 15:
    • Lecture 60 (Groebner bases V: Buchberger's Algorithm, applications of Groebner bases to elimination ideals), December 2, 2024.


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