Uniqueness of Polynomial Ring in One Variable

Theorem

Let $\struct {R \sqbrk X, \iota, X}$ and $\struct {R \sqbrk Y, \kappa, Y}$ be polynomial rings in one variable over $R$.

Then there exists a unique ring homomorphism $f: R \sqbrk X \to R \sqbrk Y$ such that:

$f \circ \iota = \kappa$

$\map f X = Y$

and it is an isomorphism.

Using the universal property, we construct ring homomorphisms in both directions, and apply uniqueness to their compositions to show that they are mutual inverses.

The existence and uniqueness of $f$ follows from the universal property.

Likewise, there is a unique ring homomorphism $g: R \sqbrk Y \to R \sqbrk X$ such that:

$g \circ \kappa = \iota$

$\map g Y = X$

and a unique ring homomorphism $h: R \sqbrk X \to R \sqbrk X$ such that:

$h \circ \iota = \iota$

$\map X = X$

By uniqueness and Identity Mapping is Ring Homomorphism, $h = \operatorname{id}$ is the identity mapping on $R \sqbrk X$.

Again by uniqueness and Composition of Ring Homomorphisms is Ring Homomorphism, $g\circ f = \operatorname{id}_{R \sqbrk X}$.

By symmetry, $f \circ g = \operatorname{id}_{R \sqbrk Y}$.

Thus $f$ is an isomorphism.

$\blacksquare$