Polynomial Rings
Table of Contents
1 Introduction
Theorem Any polynomial ring \( R[X] \) is not a field.
Proof Take \( P(X) = X \) in \( R[X] \). Suppose this has some inverse \( Q(X) \), then:
\[ X * Q(X) = 1 \]
But then evaluation at \( X = 0 \) gives \( 1 = 0 \), a contradiction, hence it cannot be the case \( P(X) \) has an inverse, thus \( R[X] \) is not a field.
This indicates the "nicest" a polynomial ring can be is a Euclidean domain.
2 Properties
- If \( R \) is a UFD then \( R[X] \) is a UFD (nontrivial), hence \( \mathbb{R}[X][Y]... \) is a UFD.
- If \( R \) is a PID then it does not imply \( R[X] \) is a PID.
- If \( R \) is a field, then \( R[X] \) is a Euclidean domain (take the Euclidean function as the degree of the polynomial).