Polynomial Forms over Field form Principal Ideal Domain/Corollary 3
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Corollary to Polynomial Forms over Field form Principal Ideal Domain
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 unique factorization domain.
Proof
We have the result Principal Ideal Domain is Unique Factorization Domain.
The result then follows from Polynomial Forms over Field form Principal Ideal Domain.
$\blacksquare$
Sources
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 65.2$ Some properties of $F \sqbrk X$, where $F$ is a field