12-240/Homework Assignment 1: Difference between revisions
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{{In Preparation}} |
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Read appendices A through D in our textbook (with higher attention to C and D), and solve the following problems: |
Read appendices A through D in our textbook (with higher attention to C and D), and solve the following problems: |
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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. |
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# 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>. |
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# 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>: |
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## <math>\frac{1}{2i}+\frac{-2i}{5-i}</math>. |
## <math>\frac{1}{2i}+\frac{-2i}{5-i}</math>. |
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## 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? |
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# 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>.) |
# 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>.) |
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Latest revision as of 06:27, 18 September 2012
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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:
- 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.
- Prove that if and are elements of a field , then if and only if or .
- Write the following complex numbers in the form , with :
- .
- .
-
- Prove that the set (endowed with the addition and multiplication inherited from ) is a field.
- Is the set (with the same addition and multiplication) also a field?
- Let be a field containing 4 elements. Assume that . Prove that . (Hint: For example, for the first equality, show that cannot equal , , or .)
