- Instructor: Mikhail Ershov
- Office: Kerchoff 302
- e-mail: ershov
*at*virginia*dot*edu - Office hours: 3 hrs TBA and by APPOINTMENT
- Course webpage:
*http://people.virginia.edu/~mve2x/3354_Spring2015*

A note on proofs. MATH 3354 is a proof-based course. Everything we discuss in class will be rigorously proved. More importantly, you are expected not only to understand proofs, but also to learn how to construct your own proofs and how to write proofs (so that others can understand your argument). It is not expected that you have taken a proof-based course before MATH 3354. However, MATH 3354 will not include any lectures devoted specifically to proof writing; instead you will be expected to develop this skill gradually as we progress through the material. If you feel that you need a more detailed introduction to proof writing, you should consider taking MATH 3000 (Transition to Higher Mathematics) before or concurrently with MATH 3354. MATH 3000 is offered this semester and will meet MWF 10-10:50am.

A note on linear algebra. The only official prerequisite for this course is MATH 1320. However, you are strongly encouraged to take linear algebra (MATH 3351 or APMA 3080) before or concurrently with MATH 3354. The only formal linear algebra skill that will be needed in MATH 3354 is the ability to add and multiply matrices (this is covered, for instance, in Section 1.6 of our book). However, some of the mateiral in MATH 3354 is based on the ideas, which also appear in linear algebra, but in less abstract setting. Thus, having taken linear algebra may help you better understand some of the topics in MATH 3354.

week | sections | topics |

Jan 13, 15 | 2.1, 2.2 | Introduction to algebraic structures. Mathematical induction. |

Jan 20, 22 | 2.3, 2.4 | Divisibility. Greatest common divisor. |

Jan 27, 29 | 2.4, 2.5 | Unique factorization theorem. Congruences. |

Feb 3, 5 | 2.5, 2.6 | Chinese Remainder Theorem. Congruence classes. |

Feb 10, 12 | 3.1, 3.2 | Definition of a group. Examples and basic properties of groups. |

Feb 17, 19 |
First midterm, 3.3 | Subgroups. |

Feb 24, 26 | 3.3, 3.4 | Subgroups (continued). Cyclic groups |

Mar 3, 5 | 3.4, 3.5 | Orders of elements. Isomorphisms. |

Mar 17, 19 | 3.6, 4.1 | Homomorphisms. Permutation groups I. |

Mar 24, 26 | 4.4 | Cosets and Lagrange Theorem. Classification of groups of small order. |

Mar 31, Apr 2 | 4.5 | Normal subgroups. Permutations groups II. |

Apr 7, 9 | Second midterm, 4.7 | Direct sums and classification of finite abelian groups. |

Apr 14, 16 | 4.6 |
Quotient Groups. |

Apr 21, 23 | 5.1, 6.1 |
Rings and ideals. |

Apr 30 | 6.2 | Quotient rings. |

- homework: 15%
- midterms: 20% each
- final: 45%

- First midterm exam: Tue, Feb 17th
- Second midterm exam: Tue, Apr 7th
- Final exam: Tue, May 5th, 9am-12pm

- Make-ups will be given only under extreme circumstances (such as serious illness). Except for emergencies, you must obtain my permission for a make-up before the exam.
- If you miss an in-class exam or fail to submit a take-home exam
by the due date without a compelling reason, you will
be assigned the score of
**0**on that exam. - University regulations specifically prohibit early make-ups.

- Homework will be assigned weekly and will usually be due on Thursday.
- Please STAPLE your homework.
- No late homework will be accepted. However, three lowest homework scores will be dropped.
- GRADING: it will not be possible to grade all homework problems. In each assignment 3-4 selected (but not announced in advance) problems will be graded for credit.

- You are welcome (and even encouraged)
to work on homework together, but you
*must*write up solutions independently, in your own words. In particular, you**should not be consulting others during the process of writing down your solution**.

- Monday, January 26 -- Last day to ADD a course, elect the audit option, change the grading option (grade or CR/NC), or establish an independent study
- Tuesday, January 27 -- Last day to DROP a course (deletion from the transcript)
- Wednesday, March 18 -- Last day to withdraw from a course