You are Here: Home >< Maths

# A question about homomorphisms... watch

1. If R is a domain with F=Frac(R), prove that Frac(R[x]) is isomorphic to F(x).

Let be a map sending (f(x),g(x)) to f(x)/g(x). We need to show that is a ring homomorphism. Let f,g,h,k be in R[x] such that f/h and g/k is in Frac(R[x]).

We know that

Am I right so far?

Now I have to show that this is a bijection...but I'm a bit confused...

For injection: Since (f(x),g(x)) maps to f(x)/g(x)...if f(x) and g(x) are equal, then we will that f(x)/g(x) = 1. For example, f(x)/f(x) = 1 = g(x)/g(x)...but we know that f(x) is not the same as g(x) in R[x].

For surjection: Isn't it obvious that for every f(x)/g(x), there exists an f(x) and g(x) in R[x]?

2. (Original post by Artus)
If R is a domain with F=Frac(R), prove that Frac(R[x]) is isomorphic to F(x).

Let be a map sending (f(x),g(x)) to f(x)/g(x). We need to show that is a ring homomorphism.
If you note that F(x) is the field of fractions of F[X], the question becomes more straightforward. EDIT: Just to make it a bit clearer - you can prove the statement without explicitly constructing a homomorphism if you use this method.

But referring to what you've written already, presumably you're writing (f(x),g(x)) for the equivalence class of pairs of elements (a(x),b(x)) (with a(x) and b(x) elements of R[X]) such that f(x)b(x)-g(x)a(x)=0? You need to check that your map phi is well-defined - if (f(x),g(x))=(h(x),k(x)), then does f(x)/g(x)=h(x)/k(x) in F(x)?
3. (Original post by Mark13)
If you note that F(x) is the field of fractions of F[X], the question becomes more straightforward. EDIT: Just to make it a bit clearer - you can prove the statement without explicitly constructing a homomorphism if you use this method.

But referring to what you've written already, presumably you're writing (f(x),g(x)) for the equivalence class of pairs of elements (a(x),b(x)) (with a(x) and b(x) elements of R[X]) such that f(x)b(x)-g(x)a(x)=0? You need to check that your map phi is well-defined - if (f(x),g(x))=(h(x),k(x)), then does f(x)/g(x)=h(x)/k(x) in F(x)?
Actually, I did mean that F(x) was teh field of fractions of F[x]. In our textbook, they say that [a,b] is a/b...so I said that (f(x), g(x)) is f(x)/g(x)...do you think that's wrong?
4. (Original post by Artus)
Actually, I did mean that F(x) was teh field of fractions of F[x]. In our textbook, they say that [a,b] is a/b...so I said that (f(x), g(x)) is f(x)/g(x)...do you think that's wrong?
That's fine, you just need to make sure you check your map is well-defined.

If you can show that there's a subfield of Frac(R[X]) isomorphic to F, and hence a subfield isomorphic to F[X], then can you see why it must be the case that Frac(R[X]) is isomorphic to Frac(F[X])?
5. (Original post by Mark13)
That's fine, you just need to make sure you check your map is well-defined.

If you can show that there's a subfield of Frac(R[X]) isomorphic to F, and hence a subfield isomorphic to F[X], then can you see why it must be the case that Frac(R[X]) is isomorphic to Frac(F[X])?
I'm not sure, but I'm not trying to prove that Frac(R[x]) is isomorphic to Frac(F[x]), I'm trying to prove that it is isomorphic to F(x)...
6. (Original post by Artus)
I'm not sure, but I'm not trying to prove that Frac(R[x]) is isomorphic to Frac(F[x]), I'm trying to prove that it is isomorphic to F(x)...
But F(X) is Frac(F[X]).

### Related university courses

TSR Support Team

We have a brilliant team of more than 60 Support Team members looking after discussions on The Student Room, helping to make it a fun, safe and useful place to hang out.

This forum is supported by:
Updated: January 1, 2013
Today on TSR

### Exam Jam 2018

Join thousands of students this half term

Poll
Useful resources

### Maths Forum posting guidelines

Not sure where to post? Read the updated guidelines here

### How to use LaTex

Writing equations the easy way

### Study habits of A* students

Top tips from students who have already aced their exams