14-240/Homework Assignment 1 - Revision history

Boyang.wu: /* Scanned Assignment Solution by Boyang.wu */

2014-12-08T19:04:33Z

Scanned Assignment Solution by Boyang.wu

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	==Scanned Assignment ~~Solution~~ by [[User Boyang.wu\|Boyang.wu]]==		==Scanned Assignment Solutions by [[User Boyang.wu\|Boyang.wu]]==
	[[File:A11.pdf]]		[[File:A11.pdf]]
	[[File:A12.pdf]]		[[File:A12.pdf]]

Boyang.wu: /* Scanned Assignment Solution by Boyang.wu */

2014-12-08T19:04:19Z

Scanned Assignment Solution by Boyang.wu

← Older revision		Revision as of 15:04, 8 December 2014
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	==Scanned Assignment Solution by [[User Boyang.wu\|Boyang.wu]]==		==Scanned Assignment Solution by [[User Boyang.wu\|Boyang.wu]]==
			[[File:A11.pdf]]
			[[File:A12.pdf]]

Boyang.wu at 18:18, 8 December 2014

2014-12-08T18:18:08Z

← Older revision		Revision as of 14:18, 8 December 2014
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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?


			==Scanned Assignment Solution by [[User Boyang.wu\|Boyang.wu]]==

Drorbn at 16:33, 15 September 2014

2014-09-15T16:33:30Z

← Older revision		Revision as of 12:33, 15 September 2014
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	{{In Preparation}}
	{{14-240/Navigation}}		{{14-240/Navigation}}

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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>.)

Drorbn: Created page with "{{In Preparation}} {{14-240/Navigation}} This assignment is due at the tutorials on Tuesday September 23. Here and everywhere, '''neatness counts!!''' You may be brilliant an..."

2014-09-03T15:03:30Z

Created page with "{{In Preparation}} {{14-240/Navigation}} This assignment is due at the tutorials on Tuesday September 23. Here and everywhere, '''neatness counts!!''' You may be brilliant an..."

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{{In Preparation}}
{{14-240/Navigation}}

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 <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>.
# 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>(1+i)^5</math>.
#
## 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?
# 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>.)