# 14-240/Homework Assignment 1

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