Polynomials Closed under Addition
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Theorem
Polynomials over Ring
Let $\struct {R, +, \circ}$ be a ring.
Let $\struct {S, +, \circ}$ be a subring of $R$.
Then $\forall x \in R$, the set $S \sqbrk x$ of polynomials in $x$ over $S$ is closed under the operation $+$.
Polynomial Forms
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 polynomials in the indeterminates $\set {X_j: j \in J}$ over the ring $R$.
Then the operation of polynomial addition on $f$ and $g$:
Define the sum:
- $\ds f \oplus g = \sum_{k \mathop \in Z} \paren {a_k + b_k} \mathbf X^k$
Then $f \oplus g$ is a polynomial.
That is, the operation of polynomial addition is closed on the set of all polynomials on a given set of indeterminates $\set {X_j: j \in J}$.