Organization of R polynomial x by classification of R

Short, sweet, to the point.

Organization of $R[x]$ by classification of $R$

Taken from Introductory Ring Theory as it is a useful reference.

If $R$ is a commutative ring with identity:

1. then $R[x]$ is a commutative ring with identity
2. Evaluation homomorphism ${\psi }_{\alpha }:R[x]\to R$ exists defined by ${\psi }_{\alpha }(p(x))=p(\alpha )$. Called the evaluation homomorphism.

If D is an integral domain:

1. $D[x]$ is an integral domain
2. Product of polynomials in $D[x]$ has the same degree as the sum of their degrees. i,e
   $(x^2+3)(x^3+x^2+34)=x^5...$

If F is an field:

1. DOES NOT MEAN $F[x]$ IS A FIELD
2. Division algo holds so do its byproducts:
   1. $(x-\alpha )$ is a factor $⟺$ $\alpha$ is a zero
   2. Most zeros as degree of polynomial
   3. $\mathrm{\exists }$ unique $r(x)$ and $s(x)$ such that $GCD(a(x),b(x))=a(x)r(x)+b(x)s(x)$
      1. Similar to what was used in All Ideals in integers mod n
3. $F[x]$ is an UFD