Ideal of Ring of Polynomials over Field has Unique Monic Polynomial forming Principal Ideal

Theorem

Let $F$ be a field.

Let $F \sqbrk X$ be the ring of polynomials in $X$ over $F$.

Let $J$ be a non-null ideal of $F \sqbrk X$.

Then there exists exactly one monic polynomial $f \in F \sqbrk X$ such that:

$J = \ideal f$

where $\ideal f$ is the principal ideal generated by $f$ in $F \sqbrk X$.

Proof

Let $f_1$ and $f_2$ be generators of $J$.

Then $f_1$ and $f_2$ are unit multiples of each other.

The units of $F \sqbrk X$ are the non-zero elements of $F$.

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Sources

• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): Chapter $9$: Rings: Exercise $22$