Monday and Friday: 10:30-11:30 AM

Tuesday 3:30-4:30 PM

Thursday 9:30-10:30 AM

Abstract algebra is one of the fundamental branches of modern mathematics. While it has its roots as far back as the late 1700's, it first came to prominence in the early 1900's. There are many current active areas of research in algebra, and the underpinnings of internet security come from algebra. This course will introduce you to several of the primary object of study in the subject: groups, rings, and fields.

Syllabus

2nd Half Updates

Homework

Class Date |
Pages |
Section(s) Covered |
Words |
---|---|---|---|

Jan. 22 | 10-14, 16-19 | 1, 2 : Intro to Groups | binary operation, group, identity, inverse, general linear group of degree 2 |

Jan. 24 | 25-30, 34 | 3 : Basic Properties of Groups | abelian group, order of a group |

Jan. 27 | 4, 6-7, 20-21, 35- 37 | 0, 2, 4 : Integers | a divides b, prime, greatest common divisor, relatively prime, least common multiple (see exercise 4.31) |

Jan. 29 | 33-34,38 | 4 : Order of an Element | order of an element, element of finite order, element of infinite order |

Jan. 31 | 39-40 | 4 : Cyclic Groups | cyclic group, generator of a group |

Feb. 3 | notes*, 74-76 | 8, Problem #8.15 : Dihedral Group | plane symmetry, regular polygon, dihedral group of order 2n (all on attached notes) |

Feb. 5 | 43-50 | 5: Subgroups | subgroup, proper subgroup, trivial subgroup, subgroup lattice |

Feb. 7 | 50-52, 59-62 | 5, 7: Cyclic Subgroups / Functions | function, onto, one-to-one, domain, range, bijection, inverse function |

Feb. 10 | 63-64, 66-69 | 8 : Symmetric Groups | cycles, disjoint cycles, symmetric group on X, transposition |

Feb. 12 | 70-73 | 8 : Alternating Group | even permutation, odd permutation, alternating group of degree n |

Feb. 14 | 81-83 | 9: Equivalence Relations and Z/nZ | equivalence relation, equivalence class |

Feb. 17 | 21-22 | 2: Z/nZ, U(n) | remainder of a mod n, congruent modulo n, additive group of integers mod n |

Feb. 19 | 83-85 | 9: Cosets | right coset of H in G, left coset |

Feb. 21 | 83-85, 88-91 | 9, 10 : Cosets, Lagrange's Theorem | Lagrange's Theorem, cardinality, index, coset representative (see 82) |

Feb. 24 | 92,103-104 (assuming abelian) | 11 : Quotient Group | quotient of abelian groups |

Feb. 26 | 99-102 | 11: Normal Subgroups, Quotient Groups | normal subgroups, quotient groups |

Feb. 28 | 103-105, 55-57 | 11 : Cauchy's Theorem and other Consequences | |

Mar. 2 | 55-57, 109-113 | 6,12 : Direct Product, Homomorphisms | direct product, homomorphism, isomorphism, monomorphism, epimorphism, automorphism |

Mar. 4 | Exam 1 | 1-5, 7-10 | |

Mar. 6 | 114-115, 121-122 | 12, 13 : Homomorphism Properties | inverse image, homomorphic image |

Mar. 9 | 122-124, 127 | 13: Isomorphism Theorems | kernel, fundamental theorem on gp homomorphisms, |

Mar. 11 | 125-126, 128-130, 153-154 | 13, 16 : Isom. Theorems, Rings | second and third isomorphism theorems, ring, commutative ring, unity |

Mar. 13 | 155-160 | 16: Ring Properties, Integral Domains | zero divisor, unit , nilpotent, trivial ring, integral domain, division ring, field |

Mar. 30/31 | 164-170 | 17: Subrings, Ideals, Prime Ideals | subring, ideal, proper ideal, trivial ideal, principal ideal generated by a |

Apr. 1/2 | 171-182 | 17: Maximal Ideals, 18: Ring Homomorphisms | ring homomorphisms, canonical homomorphism, characteristic 0, characteristic p |

Apr. 6/7 | 183-193 | 18: Quotient Fields | quotient field of an integral domain, extension, polynomial rings, leading coefficient, constant polynomial |

Apr. 8/9 | 194-198 | 19: Factoring | root, irreducible, primitive |

Apr. 13/14 | 199-201 | 19: Irreducibility Tests | Gauss' Lemma, Eisenstein's Criterion, mod p Irreducibility (Thm 19.12) |

Apr. 15/16 | 205-207 | 20: Ideals in Polynomial Rings | recap prinicpal ideals and maximal ideals |

Apr. 20/21 | 211-212, 217-218, 220-221 | 21: Principal Ideal Domain | prime, irreducible, associates, principal ideal domain, Euclidean domain |

Apr. 22/23 | -- | Flexible Day/Exam Prep | |

Apr. 24 | Exam 2 [instructions] | 6, 11-13, 16-19 | |

Apr. 27/28 | 213-216, 218-220 | 21: Unique Factorization Domain | unique factorization domains |

Apr. 29/30 | 93-94, 98 (10.32), Notes on Pweb: 133-139 | Introduction to Group Actions | centralizer, conjugacy classes, group acting on a set, orbit, stabilizer, not in book: faithful action, kernel, transitive action |

May 4/5 | 95, Notes on PWeb: 140-144 | The Class Equation and Consequences | |

May 6/7 | Notes on PWeb: 144-148 | Simplicity of A_{5} |

HW | Due Date | Problems |
---|---|---|

# 0 | Jan. 24 | Email me with answers from class 1/22, and read syllabus |

# 1 | Jan. 31 | [pdf] |

# 2 | Feb. 7 | [pdf] |

# 3 | Feb. 14 | [pdf] |

# 4 | Feb. 21 | [pdf] |

# 5 | Feb. 28 | [pdf] (last updated: 2/25) |

# 6 | [pdf] | |

# 7 | Apr. 3 | [pdf] |

# 8 | Apr. 10 | [pdf] |

# 9 | Apr. 17 | [pdf] |

# 10 | May 1 | [pdf] |

#11 | May 8 | [pdf] |