# 06-240/Homework Assignment 1

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. 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?
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 ${\displaystyle b}$.)