If F is an infinate field, prove that the polynomial ring F[x] is isomorphic to the ring T of all polynomial functions from F to F
The Attempt at a Solution
T is isomorphic to F[x]
f(a+b) = f(a) + f(b)
It is surjective by the definition of T
Injective: If f is not equal to g in F[x], then f(x) is not equal to g(x) in T.
Let h=f-g and assume h is not 0 in F
Claim: h(x) is not equal to zero in T, then h(a)=0 for every a in F. Thus every a in F is a root of the polynomial h in F[x}. However, there is no nonzero polynomial with infinately many roots, thus h=0 in F, which contradicts the asumption of h is not 0 in F, so injective. So F[x] is isomorphic to the ring T.