# The cardinality of a integral domain and its quotient field.

Let R1、R2 be two integral domains and F1、F2 be their quotient field. Suppose F1 is isomorphic to F2, I want to know if R1 is isomorphic to R2 or |R1|=|R2|.

1. If R1 and R2 are fields, then it is trivial.

2. If only one of them is an integral domain, then it is not necessary for R1 isomorphic to R2. Ex: R1=Q R2=Z Their quotient fields are exact the same, but clearly they are not isomorphic.

3. If neither of them is a field, then we also can't say R1 is isomorphic to R2. Ex: R1={\frac{a+b*(\sqrt -3)}{2}|a,b are integers and a+b is even} R2=Z[\sqrt -3]
4. But I don't know whether |R1|=|R2|.

I think that's an interesting question to me because I find that the relationship between an integral domain and its quotient field is quite similar to a given set and the free group on it, and a basis and the free abelian group constructed by it. And all these have a certain "mapping property"(ex. free object on groups).In the free (abelian) group, we have this theorem:

Let X、Y be two sets, G1、G2 are two free groups on it, then |X|=|Y| if and only if G1 is isomorphic to G2.

So I think if the preceding 4 is true, we can get a similar conclusion about an integral domain and its quotient field.

Many thanks!

If $$R$$ is an integral domain and $$F$$ is its field of fractions, then $$|R|=|F|$$. Indeed, if $$R$$ is finite then it must be a field, so $$R\cong F$$ and $$|R|=|F|$$ trivially. If $$R$$ is infinite, then since every element of $$F$$ is a fraction of two elements of $$R$$, we have $$|F|\leq |R|^2=|R|$$. But of course $$R$$ naturally embeds in $$F$$ so $$|R|\leq|F|$$ as well, and so $$|F|=|R|$$.