12-240/Homework Assignment 1

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!

This assignment is due at the tutorials on Thursday September 27. 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 $ab=0$ if and only if $a=0$ or $b=0$.
3. 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. $(1+i)^{5}$.
4. 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?
5. 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 $b$.)