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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.

A polynomial in $R[x_1,\ldots,x_n]$ may also be viewed as a polynomial in $x_n$ whose coefficients are polynomials in $x_1,\ldots, x_{n-1}$. (In other words, $R[x_1,\ldots,x_{n-1}][x]=R[x_1,\ldots, x_n]$).$ In this case we can talk about the degree of a polynomial as the highest power of $x_n$ with nonzero coefficient.

A polynomial in variables $x_1,\ldots, x_n$ is homogeneous if all monomials have the same total degree. Any polynomial in $n$ variables can be written as a sum of homogeneous polynomials.

Proposition: $R[x]$ is a principal ideal domain if and only if $R$ is a field. If $R$ is a field, then $R[x]$ is a Euclidean domain.

Unique Factorization in $R[x]$.

Theorem: If $R$ is a UFD, then so is $R[x]$.

Criteria for irreducibility

  • Polynomials of degree $2$ or $3$ over a field are irreducible or have a root in the field.
  • If a monic polynomial is irreducible in $R/I[x]$, it is irreducible in $R[x]$.

Theorem: (Eisenstein’s Criterion) Let $P$ be a prime ideal of $R$ and suppose $f(x)$ is a monic polynomial in $R[x]$ of degree $n$. If $f(x)\equiv x^{n}\pmod{P}$ and the constant term $a_0$ of $f$ is not in $P^2$ then $f$ is irreducible.

If a monic polynomial with integer coefficients has all its coefficients except its leading one divisible by a prime $p$, and its constant term is divisible by $p$ but not $p^2$, then it is irreducible.

Corollary: The polynomial $f(x)=\frac{x^p-1}{x-1}$, the $p^{th}$ cyclotomic polynomial, is irreducible in $\Z[x]$ (and $\Q[x]$).