Equality of Monomials of Polynomial Ring in One Variable

Theorem

Let $R$ be a commutative ring with unity.

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

Let $k, l \in \N$ be distinct natural numbers.

Then the mononomials $X^k$ and $X^l$ are distinct, where $X^k$ denotes the $k$th power of $X$.

Proof

By:

• Uniqueness of Polynomial Ring in One Variable
• Homomorphism Preserves Indexed Products

we may assume $R \sqbrk X$ is the ring of sequences of finite support over $R$, and $X$ is the sequence $\sequence {0, 1, 0, 0 \ldots}$.

One verifies that, for $k \ge 0$, $X^k$ is the sequence with $\map {X^k} l = \delta_{k l}$, where $\delta$ is the Kronecker delta.

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Thus $X^k \ne X^l$ if $k \ne l$.

$\blacksquare$

Also see

• Equality of Monomials of Polynomial Ring