Lecture notes for Math 55a: Honors Abstract Algebra (Fall 2017)

If you find a mistake, omission, etc., please let me know by e-mail.

The orange balls mark our current location in the course, and the current problem set.

Ceci n’est pas un Math 55a syllabus
[No, you don’t have to know French to take Math 55a. Googling ceci+n'est suffices to turn up the explanation, such as it is.]
The CAs for Math 55a are Vikram Sundar (vikramsundar@college) and Rohil Prasad (prasad01@college)
[if writing from outside the Harvard network, append .college.edu to ...@harvard].
CA office hours are Monday 8-10 PM in the Leverett Dining Hall, starting September 4 (same place and time that Math Night will start the week following).
Thanks to Vikram for setting up this Dropbox link for the CAs’ notes from class.

Section times:
Vikram Sundar: Monday 1-2 PM; Science Center room 112 on Sep.11, and room 222 from Sep.18 on.
Rohil Prasad: Thursday 4-5 PM, Science Center room 411
! If you are coming to class but not officially registered for Math 55 (e.g. you are auditing, or still undecided between 25a and 55a but officially signed up for 25a), send me your e-mail address so that I and the CA's can include you in class announcements.
My office hours for the week of 18-22 September will be Wednesday (Sep.20), not the usual Tuesday. (Still 7:30 to 9:00 PM in the Lowell House Dining Hall.)
Here is some more information from last year on the number 5777 etc. (converted to MathJax and with the added remark on $5779 = L_{27}/L_9$); as noted in the Sep.20 lecture, the fact that the palindrome 5775 factors so smoothly ($3 \cdot 5^2 \cdot 7 \cdot 11$) is also due in part to the fact that $5776 = 76^2$. Shanah Tovah!
! The diagnostic quiz will be given Wednesday, September 27 in class (11:07 AM to 12:00 noon). It will cover only material from the first three problem sets.

August 30: “Math blackboard” ($\rm\TeX$’s \mathbb font), such as $\mathbb R$, is a printed representation of a handwritten representation of ordinary boldface such as $\bf R$. When using $\rm\TeX$ (or $\rm\LaTeX$ etc.), you might as well use normal boldface. Either $\mathbf R$ or $\mathbb R$ means the set of real numbers, whether considered as a field, abelian group, metric space (more on this in Math 55b), or whatever other structure is relevant. Likewise $\mathbf C$ = $\mathbb C$ = the set of complex numbers; $\mathbf Q$ = $\mathbb Q$ = the set of rational numbers (quotients of integers — since the initial letter of “rational(s)” is preempted by the use of $\bf R$ for the reals); $\mathbf Z$ = $\mathbb Z$ = the set of integers (from German Zahlen); and in Axler, $\mathbf F$ = $\mathbb F$ = the field $\bf R$ or $\bf C$.

At least in the beginning of the linear algebra unit, we’ll be following the Axler textbook closely enough that supplementary lecture notes should not be needed. Some important extensions/modifications to the treatment in Axler:

While the third edition of Axler includes quotients and duality, it still lacks tensor algebra. This is no surprise, but it will not stop us in Math 55! Here’s an introduction [As you might guess from \oplus, the TeXism for the tensor-product symbol is \otimes.]
Corrected 14.x.2017 [Alec Sun]: at the end of the first display on page 2, it’s $w_{ij}$, not $u_i \otimes v_j$.
Thanks to Vikram for this $\rm\LaTeX$ template for problem-set solutions (here’s what the resulting PDF looks like). They ask that e-mail submissions of problem sets have “Math 55 homework” in the Subject line.

First problem set / Linear Algebra I: vector space basics; an introduction to convolution rings
• “Which if any of these basic results would fail if $\bf F$ were replaced by $\bf Z$?” — but don’t worry about this for problems 7 and 24, which specify $\bf R$.
• Problem 12: If you see how to compute this efficiently but not what this has to do with Problem 8, please keep looking for the connection.
Here’s the “Proof of Concept” mini-crossword with links concerning the ∎ symbol. Here’s an excessively annotated solution.

Second problem set / Linear Algebra II: dimension of vector spaces; torsion groups/modules and divisible groups
About Problem 5: You may wonder: if not determinants, what can you use? See Axler, Chapter 4, namely 4.8 through 4.12 (pages 121–123), and note that the proof of 4.8 (using techniques we won’t cover till next week) can be replaced by the ordinary algorithm for polynomial long division, which you probably learned with real coefficients but works over any field. While I’m at it, 4.7 (page 120) works over any infinite field; Axler’s proof is special to the real and complex numbers, but 4.12 yields the result in general. (We already remarked that this result does not hold for finite fields.)

Third problem set / Linear Algebra III: Countable vs. uncountable dimension of vector spaces; linear transformations and duality
corrected 18 September (Mark Kong):
• Problem 2: Suppose that for some (finite) $n$ we can extend $B_0$ by $n$ vectors (not “extend $B$ by $n$ vectors” etc.).
• Also, in Problem 1 Mark notes that one already needs a bit of the Axiom of Choice even to prove the fact (which I blithely asserted in class) that a countable union of countable, or even finite, sets is itself countable. (If you can enumerate a countable disjoint union $\bigcup_{i=1}^\infty S_i$ of countable or finite sets, then you can choose an element of $\prod_{i=1}^\infty S_i$ by choosing from each $S_i$ the element that comes earliest in the enumeration.) Go ahead and assume this for Problem 1.
• (And in Problem 10 it’s subsets of fewer than $e$ elements of $F$, not $e$-element subsets — but that’s still not polynomial in $q$.)

Fourth problem set / Linear Algebra IV: Duality, and connections with projective spaces and with vector spaces of polynomials
corrected 27.ix.2017: In problem 2ii, we need nonzero $x \in F$ such that $x^n \neq 1$ (not “$x^n=1$” which always exists);
and the introductory sentence now makes explicit the intention that $F$ is a finite field of $q$ elements also for problem 3.
(Noted by Forrest Flesher)

Fifth problem set / Linear Algebra V: “Eigenstuff” (preceded by prelude: exact sequences and more duality)
corrected 8.x.2017: CJ Dowd is the first to note that in problem 9 (Axler 5A:31) we cannot quite let $\bf F$ be arbitrary:
if it is finite and of size less than $m$ then it cannot contain enough pairwise distinct eigenvalues to accommodate $v_i$
for each $i=1,2,\ldots,m$ ! Fortunately this is the only obstruction, so for this problem assume that $\bf F$ contains
at least $m$ distinct elements.

Sixth problem set / Linear Algebra VI: $\bigotimes$ (and also eigenstuff cont’d, and a bit on inner products)