# Field of quotients

1. Feb 1, 2008

### ehrenfest

1. The problem statement, all variables and given/known data
Describe the field of quotients of the integral subdomain D = {n+mi|n,m in Z} of the field of complex numbers. "Describe" means give the elements of C that make the field of quotients of D in C.

2. Relevant equations

3. The attempt at a solution
So any complex number that has the form (nn'+mm'+i(nm'+mn'))/(n'^2+m'^2) will be in the field...but how can I be more descriptive...

Last edited: Feb 1, 2008
2. Feb 1, 2008

### NateTG

Do you mean a set generators of $$D$$?

3. Feb 1, 2008

### ehrenfest

I stated the problem the way it is stated in the book, but I guess a set of generators for Quot(D) would work.

4. Feb 2, 2008

### HallsofIvy

Staff Emeritus
The "field of quotients" of the sat {m + ni} where m and n are integers (the "Gaussian integers) is, by definition, the set of things of the form (m+ ni)/(a+ bi) where both a and b are also integers. Multiplying numerator and denominator of the fraction by a- bi will make the denominator an integer and give us something of the form (x/p)+ (y/p)i. Looks to me like the field of integers is the set of numbers of the form r+ si where r and s are rational numbers.

5. Feb 2, 2008

### ehrenfest

That seems reasonable but I still need to prove that
$$\frac{nn'+mm'+i(nm'+mn')}{n'^2+m'^2}$$
hits every number of the form r+si, where r and s are rational numbers...