Polynomial Forms over Field form Integral Domain/Formulation 2

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

Let $\mathbb F$ be the set of all polynomial over $\left({F, +, \circ}\right)$ defined as sequences.

Let polynomial addition and polynomial multiplication be defined as:


 * $\forall f = \left \langle {a_k}\right \rangle = \left({a_0, a_1, a_2, \ldots}\right), g = \left \langle {b_k}\right \rangle = \left({b_0, b_1, b_2, \ldots}\right) \in \mathbb F$:
 * $f \oplus g := \left({a_0 + b_0, a_1 + b_1, a_2 + b_2, \ldots}\right)$
 * $f \otimes g := \left({c_0, c_1, c_2, \ldots}\right)$ where $\displaystyle c_i = \sum_{j \mathop + k \mathop = i} a_j b_k$

Then $\left({\mathbb F, \oplus, \otimes}\right)$ is an integral domain.