14-240/Homework Assignment 1

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 23. 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 appendices A through D in our textbook (with higher attention to C and D), and solve the following problems:

Suppose $a$ and $b$ are nonzero elements of a field $F$ . Using only the field axioms, prove that $a^{-1}b^{-1}$ is a multiplicative inverse of $ab$ . State which axioms are used in your proof.
Prove that if $a$ and $b$ are elements of a field $F$ , then $a^{2}=b^{2}$ if and only if $a=b$ or $a=-b$ .
Let $F_{4}=\{0,1,a,b\}$ be a field containing 4 elements. Assume that $1+1=0$ . Prove that $b=a^{-1}=a^{2}=a+1$ . (Hint: For example, for the first equality, show that $a\cdot b$ cannot equal $0$ , $a$ , or $b$ .)
Write the following complex numbers in the form $a+ib$ $a+ib$ , with $a,b\in {\mathbb {R} }$ $a,b\in {\mathbb {R} }$ :
1. ${\frac {1}{2i}}+{\frac {-2i}{5-i}}$ .
2. $(1+i)^{5}$ .
1. Prove that the set $F_{1}=\{a+b{\sqrt {3}}:a,b\in {\mathbb {Q} }\}$ (endowed with the addition and multiplication inherited from ${\mathbb {R} }$ ) is a field.
2. Is the set $F_{2}=\{a+b{\sqrt {3}}:a,b\in {\mathbb {Z} }\}$ (with the same addition and multiplication) also a field?