Polynomial Forms over Field form Principal Ideal Domain/Corollary 1
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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 $F$.
Let $f$ be an irreducible element of $F \sqbrk X$.
Then $F \sqbrk X / \ideal f$ is a field, where $\ideal f$ denotes the ideal generated by $f$.
Proof
It follows from Principal Ideal of Principal Ideal Domain is of Irreducible Element iff Maximal that $\ideal f$ is maximal for irreducible $f$.
Therefore by Maximal Ideal iff Quotient Ring is Field, $F \sqbrk X / \ideal f$ is a field.
$\blacksquare$
Sources
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 65.3$ Some properties of $F \sqbrk X$, where $F$ is a field