**Math 55a Honors Abstract Algebra
Fall 2015**

Place: Science Center 222

Time: Tuesdays, Thursdays 2:30 p.m. -
4 p.m.

**INSTRUCTOR: Professor
Yum-Tong Siu ** (e-mail:
**siu@math.harvard.edu**)

**Office Hours: Tuesdays, Thursdays, 11 a.m. -
12:30 p.m., Science Center
511**

**COURSE ASSISTANTS:**

**Calvin Deng**** (e-mail:
calvindeng@college.harvard.edu)**

Office Hours: Sunday 3 - 4 p.m., Mathematics Department Lounge

Office Hours: Saturday 3 - 4 p.m., Leverett Dining Hall

Syllabus: This is the first term of Math 55 which is a one-year course. Though the course title of Math 55a is "Honors Abstract Algebra" and that of Math 55b is "Honors Real and Complex Analysis", the division of the topics between Math 55a and Math 55b is not that strict. There will be material in both algebra and analysis in both Math 55a and Math 55b. No textbooks will be used for Math 55a and Math 55b. Instead, lecture notes will be posted on this course website.

FINAL GRADE COMPUTATION AND TOPICS PLANNED TO BE COVERED IN THE COURSE

LECTURE NOTES: Peano's Five Axioms (Part 1)

HOMEWORK: Homework assigned on September 3, 2015 due September 15, 2015 (LaTeX file)

LaTeX FILE AND
PDF FILE OF TEMPLATE FOR TeX*ing* HOMEWORK SOLUTIONS

Please e-mail your homework to Calvin Deng (e-mail:
calvindeng@college.harvard.edu) and Vikram Sundar (e-mail: vikramsundar@college.harvard.edu) |

LECTURE NOTES: Fundamental Theorem of Algebra from Mean Value Property of Rational Function Outside the Zero-Set of Its Denominator

LECTURE NOTES: Peano's Five Axioms (Complete)

HOMEWORK: Homework assigned on September 8, 2015 due September 15, 2015 (LaTeX file)

LECTURE NOTES: From Solving Cubic and Quartic Equations to Introduction of Groups (Part 1)

HOMEWORK: Homework assigned on September 10, 2015 due September 15, 2015 (LaTeX file)

Original Paper of Joseph Louis Lagrange of 1770 on the method of partial symmetry to solve polynomial equations. Linear transformations of three variables on p.6 and four variables on p.8 of the posted lecture notes ``From Solving Cubic and Quartic Equations to the Introduction of Groups'' are found on p.221 and p.269 respectively of the 1770 paper of Lagrange.

LECTURE NOTES: From Solving Cubic and Quartic Equations to Introduction of Groups (Complete)

HOMEWORK: Homework assigned on September 15, 2015 due September 22, 2015 (LaTeX file)

LECTURE NOTES: Review of Basic Matrix Theory

HOMEWORK: Homework assigned on September 17, 2015 due September 22, 2015 (LaTeX file)

LECTURE NOTES: Tensor Product and Exterior Product After Review of Basic Vector Space Theory (Part 1)

HOMEWORK: Homework assigned on September 22, 2015 due September 29, 2015 (LaTeX file)

LECTURE NOTES: Tensor Product and Exterior Product After Review of Basic Vector Space Theory (Parts 1 and 2)

HOMEWORK: Homework assigned on September 24, 2015 due September 29, 2015 (LaTeX file)

HOMEWORK: Homework assigned on September 29, 2015 due October 6, 2015 (LaTeX file)

LECTURE NOTES: Tensor Product and Exterior Product After Review of Basic Vector Space Theory (Parts 1, 2, and 3)

HOMEWORK: Homework assigned on October 1, 2015 due October 6, 2015 (LaTeX file)

LECTURE NOTES: Matrix Normal Form by Technique of Elementary Row and Column Operations over a Ring (Part 1)

HOMEWORK: Homework assigned on October 6, 2015 due October 13, 2015 (LaTeX file)

LECTURE NOTES: Matrix Normal Form by Technique of Elementary Row and Column Operations over a Ring (Parts 1 and 2)

LECTURE NOTES: Matrix Normal Form by Technique of Elementary Row and Column Operations over a Ring (Parts 1, 2, and 3)

HOMEWORK: Homework assigned on October 8, 2015 due October 13, 2015 (LaTeX file)

LECTURE NOTES: Matrix Normal Form by Technique of Elementary Row and Column Operations over a Ring (Parts 1, 2, 3, and 4)

In-Class Mid-Term Test on November 3, 2015 (Tuesday) |

LECTURE NOTES: Galois Theory (Part 1)

HOMEWORK: Homework assigned on October 13, 2015 due October 20, 2015 (LaTeX file)

HOMEWORK: Homework assigned on October 15, 2015 due October 20, 2015 (LaTeX file)

LECTURE NOTES: Galois Theory (Parts 1 and 2)

LECTURE NOTES: Galois Theory (Parts 1, 2, and 3)

HOMEWORK: Homework assigned on October 20, 2015 due October 27, 2015 (LaTeX file)

LECTURE NOTES: Galois Theory (Parts 1, 2, 3, and 4)

HOMEWORK: Homework assigned on October 22, 2015 due October 27, 2015 (LaTeX file)

LECTURE NOTES: Galois Theory (Parts 1, 2, 3, 4, and 5)

HOMEWORK: Homework assigned on October 28, 2015 due November 5, 2015 (LaTeX file)

SOLUTIONS OF PROBLEMS IN MID-TERM TEST OF NOVERMBER 3, 2015

LECTURE NOTES: Construction of Regular Polygon of 17 Sides by Straight Edge and Compass

HOMEWORK: Homework assigned on November 5, 2015 due November 10, 2015 (LaTeX file)

LECTURE NOTES: Lefschetz Deomposition

HOMEWORK: Homework assigned on November 10, 2015 due November 17, 2015 (LaTeX file)

HOMEWORK: Homework assigned on November 12, 2015 due November 17, 2015 (LaTeX file)

LECTURE NOTES: Composition of Space Rotations, Quaternions and Special Unitary Group of Degree Two (Part 1 Only)

HOMEWORK: Homework assigned on November 17, 2015 due November 24, 2015 (LaTeX file)

HOMEWORK: Homework assigned on November 19, 2015 due November 24, 2015 (LaTeX file)

LECTURE NOTES: Schur's Lemma, Representation of Finite Groups and Young Diagrams (Part 1 Only)

HOMEWORK: Last homework assignment assigned on November 24, 2015 due December 1, 2015 (LaTeX file)