Monomials of Polynomial Ring are Linearly Independent

Theorem

One Variable

Let $R$ be a commutative ring with unity.

Let $R \sqbrk X$ be a polynomial ring in one variable $X$ over $R$.

Then the set of monomials $\set {X^k : k \in \N}$ is linearly independent.

Multiple Variables

Monomials of Polynomial Ring are Linearly Independent/Multiple Variables

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Also see

Weaker results

• Equality of Monomials of Polynomial Ring

Stronger results

• Monomials form Basis of Polynomial Ring