- #1

Firepanda

- 430

- 0

## Homework Statement

Prove that the ring R of polynomials with real coefficients (i.e. f(x) = a

_{0}+ a

_{1}x + ... + a

_{n}x

^{n}, a

_{i}real, are elements of R) has only the constant term a

_{0}as the group of units, providing the constant term isn't zero.

## Homework Equations

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) = a

_{0}+ a

_{1}x + ... + a

_{n}x

^{n}is invertible.

=> there must exist a g(x) = b

_{0}+ b

_{1}x + ... + b

_{n}x

^{n}

such that f(x).g(x) = 1

f(x).g(x) = b

_{0}(a

_{0}+ a

_{1}x + ...) + b

_{1}x(a

_{0}+ a

_{1}x + ...) + b

_{2}x

^{2}(a

_{0}+ a

_{1}x + ...) + ... = 1

Not too sure where to go from here.. Sorry if my notation is a little off but hopefully you can understand

Thanks