14-240/Homework Assignment 2

DBN: Classes: 2014-15: 14-240 / Navigation

#	Week of...	Notes and Links
Welcome to Math 240! (additions to this web site no longer count towards good deed points)
1	Sep 8	About This Class, What is this class about? (PDF, HTML), Monday, Wednesday
2	Sep 15	HW1, Monday, Wednesday, TheComplexField.pdf,HW1_solutions.pdf
3	Sep 22	HW2, Class Photo, Monday, Wednesday, HW2_solutions.pdf
4	Sep 29	HW3, Wednesday, Tutorial, HW3_solutions.pdf
5	Oct 6	HW4, Monday, Wednesday, Tutorial, HW4_solutions.pdf
6	Oct 13	No Monday class (Thanksgiving), Wednesday, Tutorial
7	Oct 20	HW5, Term Test at tutorials on Tuesday, Wednesday
8	Oct 27	HW6, Monday, Why LinAlg?, Wednesday, Tutorial
9	Nov 3	Monday is the last day to drop this class, HW7, Monday, Wednesday, Tutorial
10	Nov 10	HW8, Monday, Tutorial
11	Nov 17	Monday-Tuesday is UofT November break
12	Nov 24	HW9
13	Dec 1	Wednesday is a "makeup Monday"! End-of-Course Schedule, Tutorial
F	Dec 8	The Final Exam
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This assignment is due at the tutorials on Tuesday September 30, or at the Math Aid Centre, Sydney Smith room 1071, at the appropriately labeled mailboxes, by Friday October 3 at 5PM. 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 1.4 in our textbook, and solve the following problems:

Problems 3a and 3bcd on page 6, problems 1, 7, 18, 19 and 21 on pages 12-16 and problems 8, 9, 11 and 19 on pages 20-21. You need to submit only the underlined problems.

Note that the numbers $1^{6}-1=0$ $1^{6}-1=0$ , $2^{6}-1=63$ $2^{6}-1=63$ , $3^{6}-1=728$ $3^{6}-1=728$ , $4^{6}-1=4,095$ $4^{6}-1=4,095$ , $5^{6}-1=15,624$ $5^{6}-1=15,624$ and $6^{6}-1=46,655$ $6^{6}-1=46,655$ are all divisible by $7$ $7$ . The following four part exercise explains that this is not a coincidence. But first, let $p$ $p$ be some odd prime number and let ${\mathbb {F} }_{p}$ ${\mathbb {F} }_{p}$ 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 $7$ ?