## Polynomial rings and unique factorization

Definition: If $R$ is an integral domain, the set of integers $n$ such that $n\cdot 1=0$ is a prime ideal in $\Z$. If this ideal is the zero ideal, $R$ has characteristic zero; if this ideal is $p\Z$ then $R$ has characteristic $p$.

Lemma: In a ring of characteristic $p$, we have $(x+y)^p=x^p+y^p$.

## Fraction fields

Suppose that $R$ is an integral domain. We can construct a field containing $R$ considering $$K(R)=\{\frac{a}{b} : a,b\in R, b\not=0\}$$ and imposing the usual “fraction rules”:

• $\frac{a}{b}+\frac{c}{d}=\frac{ad+bc}{bd}$
• $\frac{xa}{xb}=\frac{a}{b}$ if $x\not=0$.
• $\frac{a}{b}\frac{c}{d}=\frac{ac}{bd}$

See DF Section 7.5 for a more formal definition.

The fraction field $K(R)$ is the “smallest field containing $R$”.

## Polynomial rings: vocabulary and basics

Let $R$ be a commutative ring with unity.

1. An element $f\in R[x]$ is monic if its highest degree coefficient is $1$.
2. The units in $R[x]$ are the units in $R$.
3. If $R$ is an integral domain, so is $R[x]$ (look at highest degree terms of the polynomials)
4. If $I$ is an ideal of $R$, then $R[x]/IR[x]$ is isomorphic to $(R/I)[x]$.
5. If $I$ is a prime ideal in $R$, then $IR[x]$ is a prime ideal in $R[x]$.
6. If $f$ is a monic polynomial in $R[x]$ and $g$ is any polynomial, then there is a division algorithm yielding $g=qf+r$ with the degree of $r$ less than the degree of $f$.
7. If $R$ is a field, any polynomial can be made monic multiplying by the inverse of its highest degree coefficient.

The ring $R[x_1,x_2,\ldots, x_n]$ is the ring of polynomials in $n$ variables with coefficients in $R$. The terms of such a polynomial are monomials $$a(i_1,\ldots,i_n)x_1^{i_1}\cdots x_{n}^{i_{n}}.$$ The total degree of such a monomial is the sum of its degrees, and the total degree of a polynomial is the highest total degree of its monomials.