Polynomial Forms over Field form Principal Ideal Domain/Proof 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 $X$ be transcendental over $F$.
Let $F \sqbrk X$ be the ring of polynomials in $X$ over $F$.
Then $F \sqbrk X$ is a principal ideal domain.
Proof
We have that Polynomial Forms over Field is Euclidean Domain.
We also have that Euclidean Domain is Principal Ideal Domain.
Hence the result.
$\blacksquare$