Polynomial Forms over Field form Integral Domain

From ProofWiki
Jump to navigation Jump to search

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.