Polynomial Forms over Field form Integral Domain/Formulation 2

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Theorem

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 $\displaystyle c_i = \sum_{j \mathop + k \mathop = i} a_j \circ b_k$


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


Proof

As $\struct {F, +, \circ}$ is a field, it is also by definition a ring.

Thus from Polynomial Ring of Sequences is Ring we have that $\struct {\GF, \oplus, \otimes}$ is a ring.



From Field is Integral Domain, a field is also by definition an integral domain.

Let $f, g \in \GF$ such that neither $f$ nor $g$ are the null polynomial.

Let:

$\deg f = m, \deg g = n$

where $\deg$ denotes the degree of $f$ and $g$ respectively.

By Degree of Product of Polynomials over Integral Domain, the degree of $f \times g$ is $m + n$.

Then by definition of polynomial multiplication, its leading coefficient is $a_m \circ b_n$.


As by definition an integral domain has no proper zero divisors:

$a_m \circ b_n \ne 0_F$.

So, by definition, $f \otimes g$ has a leading coefficient which is not $0_F$.

That is, $f \otimes g$ is not the null polynomial

The result follows by definition of integral domain.


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