Polynomial Forms over Field form Principal Ideal Domain/Corollary 3
< Polynomial Forms over Field form Principal Ideal Domain(Redirected from Polynomial Forms over Field form Unique Factorization Domain)
Jump to navigation
Jump to search
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