5) Let \( F \) denote a field. Let \( R \) denote the set of polynomials in \( F[x] \) that have \( x \)-coefficient
zero. Note that \( R \) is a subring of \( F[x] \). Prove that \( R \) is not a UFD.
Proof. \( R = \{a_0 + a_2 x^2 + a_3 x^3 + \ldots \; | \; a_i \in F\} \). Note that \( x^2 \) is irreducible in \( R \),
because the only possible non-unit factors of \( x^2 \) are \( ax \), where \( 0 \neq a \in F \) (such \( a \) are the only units
in \( R \)). But \( ax \notin R \). The same applies to \( x^3 \), which would have to have a linear factor to be factorable.
Therefore \( x^6 = (x^2)^3 = (x^3)^2 \) has two different factorizations into irreducible elements in \( R \), which is not allowable
in UFDs, where factorization must be unique. Therefore \( R \) is not a UFD. QED.
Note: This is Ex. §9.3,#3 in Dummit & Foote.