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First off, my terminology in the title may not be correct or what others use because when searching the forums for "polynomial ring" i had a lot of trouble finding anything that matched what i meant when i coined the term....

By polynomial ring i mean, for some ring R the polynomial ring over R is the ring R[x] = {a_0 + a_1x + .... + a_nx^n + ... ; a_i is in R} under the operations (+, .)

Now to my Question -

In the case that R (as above) is a Field, F, the i understand that F[x] is an integral domain, but not a field because of the simple example that my lecturers love to throw about, x in F[x] does not have an inverse (is not a unit)

Now i understand this argument and i dont dispute it, but i was wondering if any of you could tell me, or at least start me off (as it would probably help my understanding) on how to show / prove that x has no inverse in F[x]....

I have thought about this....but im not sure the following argument is correct

suppose there was g(x) st. xg(x)=1

then xg(x)=x^0

hence g(x) = x^(0-1) = x^-1 which is not polynomial and hence not in F[x]

im thinking maybe a better argument would involve how i started but then somehow using the fact F[x] has no zero divisors would be more correct...but this is just guess work really.

sorry for the long post

cheers

Bart