Math 371 Algebra

Homework: The currently assigned homework and occasionally posted solutions to problems can be found at the homework page.


Syllabus: This course will touch on two major theories combining various topics we've covered so far:

• Representation theory: combines group theory and linear algebra to gain a better understanding of how groups (we'll be mostly concentrating on finite groups) arise as "groups of matrices." We'll cover:
  • A brief review of linear algebra: linear maps, vector space bases, change of bases, dual spaces, and bilinear forms.
    
  • The theory of semi-simple algebras: group rings and endomorphism rings.
    
  • Introduction to group representations, and the dictionary between representations and "modules for the group ring," irreducible representations, and Shur's Lemma.
    
  • Character theory: class functions, G-invariant pairings, characters of representations, orthogonality relations, character tables, and many examples.
    
  • Operations on representations: tensor product, reduced representations, inflation, induced representations, and Frobenius reciprocity.
    
  • Applications: to pure groups theory and the modern theory of particle physics.
    
  
  
• Galois theory: combines groups theory and field theory to better understand roots of polynomials equations. If we have time, we'll cover:
  • Algebraic extensions of fields: root fields and splitting fields.
    
  • Galois groups: field automorphisms, entension of automorphisms, normal extensions, the Galois group, the main theorem of Galois theory.
    
  • Applications: solvable groups and the insolvability of the general quintic equation and classical problems in geometry.

Policies
(or otherwise the small print)

Homework: I will be grading your homework and tests for this course. For general policies regarding homework, tests, grade break-down, homework lateness, illnesses, etc., please see Prof. Gerstenhaber. Homework will be assigned during Tuesday's lecture and will be due the following Tuesday, by 4 pm, in my mailbox in the mathematics department main office. Special arrangements can be made ahead of time if you're physically unable to hand in your homework (like emailing it to me if you're trapped on a desert island with internet access.)

Generally, a homework problem in this course (and in general any mathematics problem) will consist of two parts: the creative part and the write-up.

• The creative part: This is when you "solve" the problem. You stare at it, poke at it, and work on it until you understand what's being asked, and then try different ideas until you find something that works. This part is fun to do with your friends, and during this part, if you're having trouble, even in understanding what the problem's asking, you should come ask Prof. Gerstenhaber or myself for hints, either in person during office hours, or by email. This part should all be done on "scratch paper."
   
• The write-up: Now that everything about the problem is clear in your mind, you go off by yourself and write up a coherent, succinct, well-written, and grammatically correct mathematical solution on clean sheets of paper. Consider this your final draft, just as in any other course. This part you should definitely NOT do with your friends.

  This will most likely be your first "real" math course, in the sense that some problems will require an argument and not just a long calculation. In order to write an argument, you'll have to use words to convey your ideas and how they connect together. Yes, it may seem strange that in a math course you'll have to use the English language. As in any other course where you use the English language, you'll need to use it correctly, i.e. you must use complete sentences, correct grammar and spelling, etc. It's true that mixing correct English with mathematical symbols is somewhat of an art, but the author of your text, for example, provides a good example of how to do this successfully. Also, if your handwriting is illegible, then consider typing up your papers, as you would for your English class.

Links:

Mathematics Department links University of Pennsylvania Mathematics Department Blackboard Where your homework and quiz grades will be posted. Math Help Resources Math Advising Page Overview and schedules of the various department help services The Math/Maple Centers Weekday night public help session in college houses. The Tutoring Center Provides free peer-based private tutors and "cram sessions." Mathematics Department list of approved private tutors