12-240/Homework Assignment 2

DBN: Classes: 2012-13: 12-240 / Navigation Additions to this web site no longer count towards good deed points. # Week of... Notes and Links 1 Sep 10 About This Class, Tuesday, Thursday 2 Sep 17 HW1, Tuesday, Thursday, HW1 Solutions 3 Sep 24 HW2, Tuesday, Class Photo, Thursday 4 Oct 1 HW3, Tuesday, Thursday 5 Oct 8 HW4, Tuesday, Thursday 6 Oct 15 Tuesday, Thursday 7 Oct 22 HW5, Tuesday, Term Test was on Thursday. HW5 Solutions 8 Oct 29 Why LinAlg?, HW6, Tuesday, Thursday, Nov 4 is the last day to drop this class 9 Nov 5 Tuesday, Thursday 10 Nov 12 Monday-Tuesday is UofT November break, HW7, Thursday 11 Nov 19 HW8, Tuesday,Thursday 12 Nov 26 HW9, Tuesday , Thursday 13 Dec 3 Tuesday UofT Fall Semester ends Wednesday F Dec 10 The Final Exam (time, place, style, office hours times) Register of Good Deeds Add your name / see who's in!

In Preparation

The information below is preliminary and cannot be trusted! (v)

This assignment is due at the tutorials on Thursday October 4. Here and everywhere, neatness counts!! You may be brilliant and you may mean just the right things, but if the teaching assistants will be having hard time deciphering your work they will give up and assume it is wrong.

Read sections 1.1 through 1.3 in our textbook, and solve the following problems:

• Problems 3a and 3bcd on page 6, problems 1, 7, 18, 19 and 21 on pages 14-16 and problems 8, 9, 11 and 19 on pages 20-21. You need to submit only the underlined problems.

• Note that the numbers $1^{6}-1=0$, $2^{6}-1=63$, $3^{6}-1=728$, $4^{6}-1=4,095$, $5^{6}-1=15,624$ and $6^{6}-1=46,655$ are all divisible by $7$. The following four part exercise explains that this is not a coincidence. But first, let $p$ be some odd prime number and let ${\mathbb {F} }_{p}$ be the field with p elements as defined in class.
  1. Prove that the product $b:=1\cdot 2\cdot \ldots \cdot (p-2)\cdot (p-1)$ is a non-zero element of ${\mathbb {F} }_{p}$.
  2. Let $a$ be a non-zero element of ${\mathbb {F} }_{p}$. Prove that the sets $\{1,2,\ldots ,(p-1)\}$ and $\{1a,2a,\ldots ,(p-1)a\}$ are the same (though their elements may be listed here in a different order).
  3. With $a$ and $b$ as in the previous two parts, show that $ba^{p-1}=b$ in ${\mathbb {F} }_{p}$, and therefore $a^{p-1}=1$ in ${\mathbb {F} }_{p}$.
  4. How does this explain the fact that $4^{6}-1$ is divisible by $7$?

You don't need to submit this exercise at all, but you will learn a lot by doing it!