# Polynomial Forms over Field form Integral Domain

## Theorem

### Formulation 1

Let $\left({F, +, \circ}\right)$ be a field whose zero is $0_F$ and whose unity is $1_F$.

Let $X$ be transcendental in $F$.

Let $F \left[{X}\right]$ be the ring of polynomial forms in $X$ over $F$.

Then $F \left[{X}\right]$ 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 $\Bbb F$ be the set of all polynomials over $\struct {F, +, \circ}$ defined as sequences.

Let polynomial addition and polynomial multiplication be defined as:

$\forall f = \sequence {a_k} = \tuple {a_0, a_1, a_2, \ldots}, g = \sequence {b_k} = \tuple {b_0, b_1, b_2, \ldots} \in \Bbb F$:
$f \oplus g := \tuple {a_0 + b_0, a_1 + b_1, a_2 + b_2, \ldots}$
$f \otimes g := \tuple {c_0, c_1, c_2, \ldots}$ where $\displaystyle c_i = \sum_{j \mathop + k \mathop = i} a_j \circ b_k$

Then $\struct {\Bbb F, \oplus, \otimes}$ is an integral domain.