# 12-240/Homework Assignment 1

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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 $b$.)