The official prerequisite for this course is Math 54 or equivalent preparation in linear algebra. It is also helpful to have previous experience with logic and proofs, sets, and mathematical induction, for instance as covered in Math 55.
You will find some review material on logic, sets, induction, complex numbers and linear algebra in Appendices A-E of the course textbook.
Exam dates:
All exams take place during the usual two-hour class period. You will need to bring your own `blue books' and scratch paper. The final exam will cover all course material but with extra emphasis on topics not yet covered on midterm exams.
Exam policies: you may consult handwritten notes which you prepare in advance. No other books, printed materials, or electronic devices may be used.
In general there will be no make-up exams and no dropped scores. In case of illness or emergency, I will consider making allowances for missed exams on an individual basis, provided you have a valid reason and contact me as soon as possible.
Homework policies: usually two or three problems from each homework set will be chosen for grading.
You may work with others to solve homework problems, but you must write your solutions indiviually. Copying solutions from other students, classes in previous years, the internet, or elsewhere is not allowed.
You can view your homework and exam scores on the class bCourses page (choose "Grades" from the bCourses menu). I will also use bCourses to post homework solutions.
All homework is due on the Thursday of the week for which it is assigned.
Chapter numbers in reading assignments refer to Goodman. I also plan to distribute supplementary notes on Euler's function φ(n), geometric constructions, finite fields, and the Galois group of a general polyomial.
Week | Topics | Reading | Homework |
---|---|---|---|
June 22-25 | Groups of symmetries and permutations. Divisibility, GCD, prime factorization. Modular arithmetic. | 1.1-1.7 | Problem Set 1, Solutions^{*} |
June 29-July 2 | Chinese remainder theorem. Euler's function φ(n). Groups, subgroups and cyclic groups. Dihedral groups. Homomorphisms and normal subgroups. Cosets, Lagrange's theorem, order and index of subgroups. Quotient groups. | 1.10, 2.1-2.5, 2.7, Notes on φ(n) | Problem Set 2, Solutions |
July 6-9 | Subgroups of G/N and Z_{n}. Homomorphism theorems. Partitions and equivalence relations. Direct products. | 2.6-2.7, 3.1 | Problem Set 3, Solutions.
— Midterm 1, Solutions |
July 13-16 | Semidirect products. Smith normal form and structure of finitely generated abelian groups. Symmetries of regular polyhera. Group actions, orbits, stabilizers; Burnside's lemma. Automorphisms, class equation, solvability of p-groups. | 3.2, 3.5-3.6, 4.1-4.3, 5.1-5.4 (through Cor. 5.4.5), 11.1, 11.4 | Problem Set 4, Solutions |
July 20-23 | Polynomials. Rings and fields. Homomorphisms, ideals, direct sums, quotients. Integral domains, relationship between divisibility and principal ideals, PID's. Examples: fields (familiar and new), polynomial rings, the ring of integers. | 1.8, 1.11, 6.1-6.4 | Problem Set 5, Solutions |
July 27-30 | PID implies UFD. Prime and maximal ideals. Construction of fields K[x]/(p(x)) where p(x) is an irreducible polynomial. Fraction field of an integral domain. Divisibility and gcd in a UFD. Gauss's Lemma. R UFD implies R[x] UFD. Irreducibility tests. | 6.5, 6.6 through page 313, 6.8, 7.1-7.2 | Problem Set 6^{**}, Solutions.
— Midterm 2, Solutions |
Aug 3-6 | Vector spaces over a field. Field extensions, dimension, algebraic and transcendental elements. Finite and algebraic extensions. Construction of extension K(α) of K. Roots and splitting fields. Impossibility of angle trisection. Finite fields. Preview of Galois theory for splitting fields in the complex numbers. | Guide to field theory; parts of 3.3, 7.3-7.5, 9.1-9.2 | Problem Set 7^{**}, Solutions |
Aug 10-13 | Separability criteria. Definition and characterization of Galois extensions. Main theorem of Galois theory. Galois group of a general polynomial. Solvable groups, simple groups, unsolvability of the general quintic by radicals. | Notes on finite fields; parts of 9.3-9.7, 10.1-10.6
— Review Guide |
Problem Set
8, Solutions
— Final Exam, Solutions — Online Course Evaluation |
^{*}Homework problems chosen for grading:
^{**}Corrections: PS 6—omit 6.8.5(c); PS 7—9.2 3 should be 9.2 2.