Prove that the ring R of polynomials with real coefficients (i.e. f(x) = a0 + a1x + ... + anxn, ai real, are elements of R) has only the constant term a0 as the group of units, providing the constant term isn't zero.
u is a unit if there exists a v such that uv=1
The Attempt at a Solution
Clearly all the constant terms are units of the ring as we are dealing with the real numbers.
Claim: There are more than just these, and that a polynomial f(x) = a0 + a1x + ... + anxn is invertible.
=> there must exist a g(x) = b0 + b1x + ... + bnxn
such that f(x).g(x) = 1
f(x).g(x) = b0(a0 + a1x + ...) + b1x(a0 + a1x + ...) + b2x2(a0 + a1x + ...) + ... = 1
Not too sure where to go from here.. Sorry if my notation is a little off but hopefully you can understand