Determine Q(D) for D={m+ni|m,n in Z} contained in C
Note: D is an integral domain and Q(D) is its quotient field
I know for Q(D), we have [a,b]+[c,d]=[ad+bc,bd] and [a,b]*[c,d]=[ac,bd]
I just don't know how to start this.
I was given to this hint to try.
Q(D) ∼= Q(i) = {r + si | r, s ∈ Q} by showing that the
map f : Q(i) → Q(D) given by
f(a/b+c/di) = [ad + bci, bd]
We want to show 1-1, onto, f(a+b)=f(a)+f(b) an f(ab)=f(a)f(b)
1-1:Assume a/b+c/di=e/f+g/hi
f(a/b+c/di)=f(e/f+g/hi)
[ad+bci,bd]=[eg+fgi,fh]
onto:I always get hung up onto.
f(a+b)=f((a/b+c/di)+(e/f+g/hi))=f(af+be/bf+chi+gdi/-dh)=[(af+be)(-dh)+bf(chi+gdi)(-dh)]
f(a*b):
Consider f((a/b+c/di)(e/f+g/hi))=f(ae/bf+ga/hi+ce/di+cg/-hd)
This is where I've gotten so far
Honestly, what you have written is really hard to follow.
We are trying to determine the quotient field of $\displaystyle \mathbb{Z}[i]$ (what you have written as $\displaystyle D$).
First, notice that $\displaystyle \mathbb{Z}\subseteq \mathbb{Z}[i]$, so that $\displaystyle Q( \mathbb{Z})\subseteq Q(\mathbb{Z}[i])$. Since $\displaystyle \mathbb{Q}$ is the quotient field of $\displaystyle \mathbb{Z}$, this means $\displaystyle \mathbb{Q}\subseteq Q(\mathbb{Z}[i])$.
Obviously $\displaystyle i\in Q(\mathbb{Z}[i])$. This and the previous containment force $\displaystyle \mathbb{Q}(i)\subseteq Q(\mathbb{Z}[i])$ (if $\displaystyle \mathbb{Q},i$ are both inside of the field $\displaystyle Q(\mathbb{Z}[i])$, then so is the smallest field containing $\displaystyle \mathbb{Q}$ and $\displaystyle i$).
But then $\displaystyle \mathbb{Q}(i)$ is a field satisfying $\displaystyle \mathbb{Z}[i]\subseteq \mathbb{Q}(i) \subseteq Q(\mathbb{Z}[i])$. Since the quotient field of an integral domain is the smallest field containing that integral domain, this forces equality: $\displaystyle \mathbb{Q}(i)=Q(\mathbb{Z}[i])$.