14-240/Homework Assignment 1

DBN: Classes: 2014-15: 14-240 / Navigation Welcome to Math 240! (additions to this web site no longer count towards good deed points) # Week of... Notes and Links 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 Register of Good Deeds Add your name / see who's in!

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:

1. 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.
2. Prove that if $a$ and $b$ are elements of a field $F$, then $a^{2}=b^{2}$ if and only if $a=b$ or $a=-b$.
3. 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 $b$.)
4. Write the following complex numbers in the form $a+ib$, with $a,b\in {\mathbb {R} }$:
   1. ${\frac {1}{2i}}+{\frac {-2i}{5-i}}$.
   2. $(1+i)^{5}$.
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?