# Polynomial Forms over Field form Principal Ideal Domain/Corollary 1

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 $F$.

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