Epimorphism from Polynomial Forms to Polynomial Functions

Theorem

Let $D$ be an integral domain.

Let $D \sqbrk X$ be the ring of polynomial forms in $X$ over $D$.

Let $\map P D$ be the ring of polynomial functions over $D$.

The mapping $\kappa: D \sqbrk X \to \map P D$ given by:

$\ds \map \kappa {\sum_{k \mathop = 0}^n {a_k \circ X^k} } = f$

where $\ds f = \sum_{k \mathop = 0}^n {a_k \circ x^k}, x \in D$

is a ring epimorphism.

Proof

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Sources

• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 64$ Polynomial rings over an integral domain