- #1

- 247

- 0

## Homework Statement

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

## Homework Equations

## The Attempt at a Solution

Let [tex]\phi : Frac(R[x]) \rightarrow F(x)[/tex] be a map sending (f(x),g(x)) to f(x)/g(x). We need to show that [tex]\phi[/tex] 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

[tex]\phi(1,1) = 1/1= 1[/tex]

[tex]\phi (fg, hk) = \frac{fg}{hk} = \frac{f}{h}\frac{g}{k} = \phi(f,h)\phi(g,k)[/tex]

But I'm confused with the addition part...

[tex]\phi(f+g,h+k) = \frac{f+g}{h+k}[/tex]

[tex]\phi(f,h)+\phi(g,k) = \frac{f}{h}+\frac{g}{k} = \frac{kf+gh}{h+k}[/tex]

But now [tex]\phi(f+g,h+k) \not= \phi(f,h) + \phi(g,k)[/tex]

Can anybody help with this?

Thanks in advance