Field of Quotients of Integral Subdomain in Complex Num

[ehrenfest]

Homework Statement

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.

Homework Equations

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...

 

[NateTG]

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

 

[ehrenfest]

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

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

 

[HallsofIvy]

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.

 

[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...