Week 1: September 11-15, 2016.

Required Reading: You should read all of Appendices A-D, including things that were not covered in class. Some comments:

Appendix A. You are expected to know some basic set-theory notions from high school, although the intersections and unions of infinitely many sets may be new. Try to absorb the material on relations (even though not covered in class). In class, we also discussed cartesian products A x B; it's enough to know this for finitely many sets.

Appendix B. Not covered in class, but please read anyway.

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.

Homework: Assessment #1.

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 file for each problem.) Note that scanning is available for free at many UofT libraries.

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.

Additional Homework (not to be handed in):

Prove that the set $F=\{a+b\sqrt{2}\colon a,b\in \mathbb{Q}\}$, with the usual operations of addition and multiplication, is a field.
Let $F=\{0,1,a,b\}$ be the field with 4 elements. Assuming that $1+1=0$ (which one can actually prove -- see homework #1), show that $$ b=a^{-1}=a^2=a+1.$$
Prove in detail that $\mathbb{Z}_k$ is not a field if $k$ is not a prime number. Hint: Suppose $k=l\cdot m$ (in $\mathbb{Z}$) with $l$ between $1$ and $k$. If $\mathbb{Z}_k$ were a field, then the element [$l$] $\in\mathbb{Z}_k$ would have to have a multiplicative inverse. Get a contradiction.
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:

The tutorials will start this coming week, September 18. I understand that a few students could not yet get into the tutorial of their choice. We expect that space will open up once a few students migrate to MAT223. Until then, perhaps just attend any tutorial (as long as seats are available).