Multiplication of Polynomials is Commutative

Theorem
Multiplication of polynomials is commutative.

Proof
Let $\struct {R, +, \circ}$ be a commutative ring with unity.

Let $\set {X_j: j \in J}$ be a set of indeterminates.

Let $Z$ be the set of all multiindices indexed by $\set {X_j: j \in J}$.

Let:


 * $\ds f = \sum_{k \mathop \in Z} a_k \mathbf X^k$


 * $\ds g = \sum_{k \mathop \in Z} b_k \mathbf X^k$

be arbitrary polynomials in the indeterminates $\set {X_j: j \in J}$ over $R$.

Then:

Therefore, $f \circ g = g \circ f$ for all polynomials $f$ and $g$.

Therefore, polynomial multiplication is commutative.