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Artusartos
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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