14-240/Homework Assignment 1: Difference between revisions

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# Suppose <math>a</math> and <math>b</math> are nonzero elements of a field <math>F</math>. Using only the field axioms, prove that <math>a^{-1}b^{-1}</math> is a multiplicative inverse of <math>ab</math>. State which axioms are used in your proof.
# Suppose <math>a</math> and <math>b</math> are nonzero elements of a field <math>F</math>. Using only the field axioms, prove that <math>a^{-1}b^{-1}</math> is a multiplicative inverse of <math>ab</math>. State which axioms are used in your proof.
# Prove that if <math>a</math> and <math>b</math> are elements of a field <math>F</math>, then <math>ab=0</math> if and only if <math>a=0</math> or <math>b=0</math>.
# Prove that if <math>a</math> and <math>b</math> are elements of a field <math>F</math>, then <math>a^2=b^2</math> if and only if <math>a=b</math> or <math>a=-b</math>.
⚫
# Let <math>F_4=\{0,1,a,b\}</math> be a field containing 4 elements. Assume that <math>1+1=0</math>. Prove that <math>b=a^{-1}=a^2=a+1</math>. (''Hint:'' For example, for the first equality, show that <math>a\cdot b</math> cannot equal <math>0</math>, <math>a</math>, or <math>b</math>.)
# Write the following complex numbers in the form <math>a+ib</math>, with <math>a,b\in{\mathbb R}</math>:
# Write the following complex numbers in the form <math>a+ib</math>, with <math>a,b\in{\mathbb R}</math>:
## <math>\frac{1}{2i}+\frac{-2i}{5-i}</math>.
## <math>\frac{1}{2i}+\frac{-2i}{5-i}</math>.
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## Prove that the set <math>F_1=\{a+b\sqrt{3}:a,b\in{\mathbb Q}\}</math> (endowed with the addition and multiplication inherited from <math>{\mathbb R}</math>) is a field.
## Prove that the set <math>F_1=\{a+b\sqrt{3}:a,b\in{\mathbb Q}\}</math> (endowed with the addition and multiplication inherited from <math>{\mathbb R}</math>) is a field.
## Is the set <math>F_2=\{a+b\sqrt{3}:a,b\in{\mathbb Z}\}</math> (with the same addition and multiplication) also a field?
## Is the set <math>F_2=\{a+b\sqrt{3}:a,b\in{\mathbb Z}\}</math> (with the same addition and multiplication) also a field?
⚫
# Let <math>F_4=\{0,1,a,b\}</math> be a field containing 4 elements. Assume that <math>1+1=0</math>. Prove that <math>b=a^{-1}=a^2=a+1</math>. (''Hint:'' For example, for the first equality, show that <math>a\cdot b</math> cannot equal <math>0</math>, <math>a</math>, or <math>b</math>.)

Revision as of 11:33, 15 September 2014

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
Class Photo
Add your name / see who's in!
14-240-Splash.png

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 and are nonzero elements of a field . Using only the field axioms, prove that is a multiplicative inverse of . State which axioms are used in your proof.
  2. Prove that if and are elements of a field , then if and only if or .
  3. Let be a field containing 4 elements. Assume that . Prove that . (Hint: For example, for the first equality, show that cannot equal , , or .)
  4. Write the following complex numbers in the form , with :
    1. .
    2. .
    1. Prove that the set (endowed with the addition and multiplication inherited from ) is a field.
    2. Is the set (with the same addition and multiplication) also a field?
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