Proving Isomorphism Between Fields of Quotients of Integral Domains

Understanding the relationship between integral domains and their corresponding fields of quotients is a fundamental concept in abstract algebra. One key aspect of this relationship is proving that if two integral domains are isomorphic, then their fields of quotients are also isomorphic. This article delves into the detailed steps required to establish this isomorphism.

1. Defining the Isomorphism

Given two integral domains ( R ) and ( S ), an isomorphism ( phi: R to S ) is a bijective ring homomorphism that preserves both addition and multiplication. Additionally, ( phi(r) eq 0 ) for all ( r in R setminus {0} ). This means that every non-zero element in ( R ) is mapped to a non-zero element in ( S ).

2. Understanding the Fields of Quotients

The field of quotients ( Q(R) ) of an integral domain ( R ) consists of equivalence classes of pairs ( (a, b) ) where ( a, b in R ) and ( b eq 0 ). Two pairs ( (a, b) ) and ( (c, d) ) are equivalent if ( ad bc ).

Similarly, the field of quotients ( Q(S) ) of an integral domain ( S ) is defined in the same way.

3. Constructing the Induced Map

Given the isomorphism ( phi: R to S ), we can define a map ( Phi: Q(R) to Q(S) ) by sending the equivalence class ( [a, b] ) in ( Q(R) ) to the equivalence class ( [phi(a), phi(b)] ) in ( Q(S) ).

4. Showing Well-Definedness of ( Phi )

To show that ( Phi ) is well-defined, we need to verify that if ( (a, b) sim (c, d) ) in ( Q(R) ) (i.e., ( ad bc )), then ( (phi(a), phi(b)) sim (phi(c), phi(d)) ) in ( Q(S) ).

Assume ( ad bc ). Applying ( phi ), we have: ( phi(a) cdot phi(d) phi(a) cdot phi(d) ) ( phi(b) cdot phi(c) phi(b) cdot phi(c) ) Thus, ( phi(a) cdot phi(d) phi(b) cdot phi(c) ), confirming that ( Phi ) is well-defined.

5. Showing ( Phi ) is a Homomorphism

To show that ( Phi ) is a homomorphism, we check the following properties:

5.1 Addition

For any equivalence classes ( [a, b] ) and ( [c, d] ) in ( Q(R) ):

[ Phi([a, b] [c, d]) Phileft[frac{ad bc}{bd}right] left[frac{phi(a) cdot phi(d) phi(b) cdot phi(c)}{phi(b) cdot phi(d)}right] ]

On the other hand:

[ Phi([a, b]) Phi([c, d]) [phi(a), phi(b)] [phi(c), phi(d)] left[frac{phi(a) cdot phi(d) phi(b) cdot phi(c)}{phi(b) cdot phi(d)}right] ]

This shows that ( Phi ) preserves addition.

5.2 Multiplication

A similar approach can be taken to show that ( Phi ) preserves multiplication.

6. Showing ( Phi ) is Bijective

Injective: If ( Phi([a, b]) Phi([c, d]) ), then:

[ [phi(a), phi(b)] [phi(c), phi(d)] ]

which implies ( phi(a) cdot phi(d) phi(b) cdot phi(c) ). Since ( phi ) is injective, this leads to ( a c ) and ( b d ).

Surjective: For any pair ( [phi(a), phi(b)] ) in ( Q(S) ), we can find a pair ( [a, b] ) in ( Q(R) ) such that ( Phi([a, b]) [phi(a), phi(b)] ).

Conclusion

Since ( Phi ) is well-defined, a homomorphism, and bijective, we conclude that the fields of quotients ( Q(R) ) and ( Q(S) ) are isomorphic. Thus, if two integral domains are isomorphic, their fields of quotients are also isomorphic.