constructing a field of fraction

Question

Let $\mathbb{Z}\left [ i \right ]=\left \{ a+bi:a,b \in \mathbb{Z} \right \}.$ Show that the field of quotient of $\mathbb{Z}\left [ i \right ]$ is ring isomorphic to $\mathbb{Q}\left [ i \right ]=\left \{ r+si: r,s \in \mathbb{Q} \right \}$

Let's try to construct the field of quotient:

The field of quotient is the smallest field of fraction contained in an integral domain.

Try: $F=\left \{ \frac{a+bi}{c+di} : a+bi, c+di \in \mathbb{Z}\left [ i \right ]\right \}$

I never liked these sort of questions involving function construction. I wish I could obtain a bit of help here to get me further.

Thanks in advance.

How can you express $\frac{1}{a+ib}$ ? Hint : think about the conjugate — Maxime Ramzi, Jul 15 '17 at 07:05
Aside: if you drop the \left and \right, you get a nicely typeset $\mathbb{Z}[i]$ from \mathbb{Z}[i]. — , Jul 15 '17 at 09:18

score 0 · Answer 1 · answered Jul 15 '17 at 07:10

Multiplying the fraction $\frac{a+bi}{c+di}$ by $1 = \frac{c-di}{c-di}$ (noting that $c-di$ is not zero because $c+di$ isn't) we obtain the equivalent fraction $$\frac{a+bi}{c+di} = \frac{ac + bd + (bc-ad)i}{c^2 + d^2} = \frac{ac+bd}{c^2 + q^2} + \frac{bc-ad}{c^2 + q^2}i \in \mathbb{Q}[i].$$

Conversely, any $\frac{a}{c} + \frac{b}{d}i \in \mathbb{Q}[i]$ may be written as $$\frac{a}{c} + \frac{b}{d}i = \frac{ad + bci}{cd}.$$

constructing a field of fraction

1 Answers1