12-240/Homework Assignment 1

From Drorbn
Revision as of 07:27, 18 September 2012 by Drorbn (Talk | contribs)

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search

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.
    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?
  4. 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.)