The Gaussian integers are the ones of the form $m + ni$ where $m,n$ are both integers. I need to show that given any two Gaussian integers $a$ and $b$, $ab = 0$ must imply that $a = 0$ or $b = 0$.
I tried to show it by contradiction. Assume we have $a \neq 0 $ and $b \neq 0$ such that $ab= 0$. Then $(m_1 + n_1i)(m_2+n_2i) = 0$, hence $(m_1m_2 – n_1n_2) + (m_1n_2 + m_2n_1)i = 0$, hence $ m_1m_2 = n_1n_2$ and $m_1n_2 = -m_2n_1$. We then multiply these two equations together to get $m_1^{2}(m_2n_2) = -n_1^{2}(m_2n_2)$ but then I can't really do the cancellation law because maybe $m_2 = 0$ or $n_2 = 0$. I am stuck. Any suggestions?
Best Answer
Yet another way: The Gaussian integers are clearly a subring of $\mathbb C$: certainly a subset, and the rules for addition and multiplication in the two rings are the same. Since $\mathbb C$ has no zero divisors, neither does the ring of Gaussian integers.