Answer to Question #138071 in Abstract Algebra for Ram

Question

Find the quotient field of integral domain {a+ib such that a,b belongs to Z}

Answer 1

Let $K$ be the quotient field of integral domain $\mathbb Z[i]=\{a+bi\ |\ a,b\in\mathbb Z\}$ and let $\mathbb Q(i)$ the field of complex numbers of the form $r+si$ with both $r$ and $s$ in $\mathbb Q$ . Let us prove that $K=\mathbb Q(i)$.

First thing we need to convince yourseld that $K\subset\mathbb Q(i)$. If we take a fraction $\frac{a+bi}{c+di}$ with $a,b,c,d\in\mathbb Z$ we can rewrite it as $\frac{a+bi}{c+di}=\frac{ac+bd}{c^2+d^2}+\frac{bc-ad}{c^2+d^2}i$, and we're done.

On the other hand, if $\frac{m}{n}+\frac{p}{q}i\in\mathbb Q(i)$, we can rewrite this as $\frac{nq+npi}{nq}$ where both numerator and denominator are Gaussian integers. Thus we also have $\mathbb Q(i)\subset K$ and we're finished.