Polynomial Forms over Field form Principal Ideal Domain/Corollary 2

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 $f \in F \sqbrk X$.

Then $\ideal f$ is a maximal ideal of $F \sqbrk X$ if and only if $f$ is irreducible.

Proof

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Sources

• 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $6$: Polynomials and Euclidean Rings: $\S 29$. Irreducible elements: Theorem $57$