Polynomial Forms over Field form Integral Domain
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Theorem
Formulation 1
Let $\struct {F, +, \circ}$ be a field whose zero is $0_F$ and whose unity is $1_F$.
Let $X$ be transcendental in $F$.
Let $F \sqbrk X$ be the ring of polynomial forms in $X$ over $F$.
Then $F \sqbrk X$ is an integral domain.
Formulation 2
Let $\struct {F, +, \circ}$ be a field whose zero is $0_F$ and whose unity is $1_F$.
Let $\GF$ be the set of all polynomials over $\struct {F, +, \circ}$ defined as sequences.
Let polynomial addition and polynomial multiplication be defined as:
- $\forall f = \sequence {a_k} = \tuple {a_0, a_1, a_2, \ldots}, g = \sequence {b_k} = \tuple {b_0, b_1, b_2, \ldots} \in \GF$:
- $f \oplus g := \tuple {a_0 + b_0, a_1 + b_1, a_2 + b_2, \ldots}$
- $f \otimes g := \tuple {c_0, c_1, c_2, \ldots}$ where $\ds c_i = \sum_{j \mathop + k \mathop = i} a_j \circ b_k$
Then $\struct {\GF, \oplus, \otimes}$ is an integral domain.