Week 1: September 11-15, 2017.

Required Reading: Appendices C and D, including things that were not covered in class. Some comments:

Appendic C. Covered in detail. In class, we also talked about the field $\mathbb{Z}_p$ for p prime (and the fact that $\mathbb{Z}_k$ is not a field when k is not prime). A reference for some of this material are these notes by David Vogan, which also talks about the field with 4 elements. (For now, skip the paragraph involving vector spaces. Also, note that $\mathbb{Z}/k\mathbb{Z}$ is just a different notation for what we denoted $\mathbb{Z}_k$. )

Appendix D. Covered in detail, but without a proof of the fundamental theorem of algebra.

My office hours are set for Thursdays, 1:30-2:30 or by email appointment. (Not Tuesdays as originally announced.) I'm also available right after Thursday lectures -- Tuesdays are not so good since I need to teach another class.

There will be a Putnam orientation session, with free pizza , on Friday, September 15, 4-6 in WB 116. For details, see here

This coming Tuesday (Sept 18) there will be a written quiz during the second half of class. This quiz does not count towards the course mark in any way, so it won't be written under formal `exam conditions'. However, the quiz will be graded by out TA's, scanned, and returned via email. The quiz will be about preliminaries (sets, functions), and possibly a question or two from high school level calculus/algebra. The purpose of this quiz is to give you an idea where you stand in comparison to your peers. You'll get the statistics of class performance, and it's up to you to draw your conclusions from the result.

Homework: Assessment #1.

Do not share your link with others, it's your personal link.

There will be 5 problems, each worth 5 points. The due date will be set for Friday, September 22, 2016 at 11 p.m. Do not leave it to the last minute to upload your solutions. Excuses along the lines of `my computer broke down', `we had a power outage' will (usually) not be accepted.

As you will see from the instructions given by crowdmark, you can hand-write the solutions (use a different page for each problem), and then scan them or take a picture to create a pdf or jpg file for uploading. (A separate set of pages for each problem.) Scanning is available for free at many UofT libraries. Also, I have learned that there are now apps for smartphones to scan documents, generating a pdf that's much better quality than a jpg picture. Make sure that whatever you upload is readable -- you can use the preview option on crowdmark for that. According to the crowdmark instructions, you can double-check and resubmit anytime before the due date.

Alternatively, you are very much encouraged to type the solutions using LaTeX . In case you're unfamiliar with LaTeX, here is a pdf document with some instructions, produced using LaTeX, and here is its source file . You can rightclick to downlad the source file, rename it (just make sure it ends on .tex), and modify it.

Some do's and dont's when writing proofs: We are ok with simple logic symbols such us $\forall,\ \exists, \Leftarrow, \Rightarrow, \Leftrightarrow$ provided that you use them properly . In case of doubt, plain words are often better. Please do not use the $\therefore$ symbol since this is very uncommon in mathematical writing. Also, some other logic symbols such as $\land$ and $\lor$ are not commonly used in math texts (outside of mathematical logic), it's better to use words. Finally, avoid the use of $\times $ for multiplications (of numbers etc) since there are too many other meanings attached to this symbol -- the cross product of vectors, the variable $x$, and so on. (Use $\cdot$ instead.)

Additional Homework (not to be handed in):

Prove in detail that $\mathbb{Z}_k$ is a field if and only if $k$ is a prime number. Hint: 1) Show first that if $k$ is not prime, then $\mathbb{Z}_k$ is not a field. (In fact, if $l$ is a natural number between $1$ and $k$, and $l$ divides $k$, show that $[l]$ has no multiplicative inverse. 2) (This part is harder.) Next, if $k$ is prime, show that for any given $l$ between $0$ and $l$ the numbers $[l],[2l],[3l],\ldots,[kl]$ are all distinct. Hence, one of them is equal to $[1]$. Use this to show that every non-zero element has a multiplicative inverse.
Let $P$ be the set of polynomial functions $p\colon \mathbb{R}\to \mathbb{R},\ \ x\mapsto p(x)$. Define addition and multiplication of polynomials by adding or multiplying the values pointwise. Is $P$ a field?

Let $R$ be the set of rational functions on a real variable $x$. The elements are thus functions $f$ that can be written as quotients of polynomials, e.g. $$ f(x)=\frac{1+x^2}{1-2x-x^4}.$$ The domain of definition of $f$ is the set of all $x$ for which the denominator is non-zero. We identify two such functions if they coincide on their domain of definition, for example, $\frac{x}{x}=1$. Is this $R$ a field?

Tutorials will start this coming week. If you couldn't get into the tutorial of your choice, please note that neither me or our TA's deal with enrolment into tutorials, and we cannot help to `get you in'! Typically, in a couple of weeks once some students migrate to MAT223, space in tutorials will free up. The assignments and tests for this course are independent of tutorials.