Field of Quotients of Integral Subdomain in Complex Num

• ehrenfest
In summary, the field of quotients of the integral subdomain D = {n+mi|n,m in Z} in the field of complex numbers is defined as the set of numbers of the form (m+ni)/(a+bi), where a and b are integers. This can be written as (x/p)+(y/p)i, where p is an integer and x and y are rational numbers. It can be proven that this field of quotients includes all numbers of the form r+si, where r and s are rational numbers.
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.

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

NateTG said:
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.

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.

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...

1. What is the field of quotients of an integral subdomain in complex numbers?

The field of quotients of an integral subdomain in complex numbers is a field that contains all possible quotients of elements in the integral subdomain. In other words, it is the smallest field that contains all the elements in the integral subdomain. It is also known as the fraction field or the field of fractions.

2. How is the field of quotients of an integral subdomain related to the complex numbers?

The field of quotients of an integral subdomain is a subset of the complex numbers. It contains all the possible quotients of elements in the integral subdomain, which are complex numbers. Additionally, the field of complex numbers is also the field of quotients of the integral domain of integers.

3. What is the significance of the field of quotients of an integral subdomain in complex numbers?

The field of quotients of an integral subdomain is significant because it allows for the division of elements in the integral subdomain, which is not possible in the integral subdomain itself. This field also helps in simplifying and generalizing certain mathematical concepts and proofs.

4. How is the field of quotients of an integral subdomain calculated?

The field of quotients of an integral subdomain is calculated by taking all the possible quotients of elements in the integral subdomain and including all the necessary closure properties such as addition, multiplication, and inverses. This process results in the smallest field that contains all the elements in the integral subdomain.

5. Can the field of quotients of an integral subdomain in complex numbers be extended to other number systems?

Yes, the concept of the field of quotients can be extended to other number systems such as rational, real, and p-adic numbers. In each case, the field of quotients is the smallest field that contains all the elements of the respective integral subdomain. However, the specific properties and construction of the field of quotients may differ depending on the number system.

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