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\begin{document}
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\title{Factorization of Monic Polynomials}
\author{William J. Heinzer}
\address{Mathematics Department, Purdue University, West Lafayette,
IN 47907-1395}
%\curraddr{Mathematics Department, Purdue University, West Lafayette,
%IN 47907-1395}
\email{heinzer@math.purdue.edu}
\author{David C. Lantz}
\address{Mathematics Department, Colgate University, Hamilton,
NY 13346-1398}
%\curraddr{Mathematics Department, Colgate University, Hamilton,
%NY 13346-1398}
\email{dlantz@mail.colgate.edu}
% \thanks will become a 1st page footnote.
\thanks{The second author is grateful for the hospitality and
support of Purdue University while this work was done.}
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\subjclass{Primary 13B25, 13G05, 13J1. }
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%\dedicatory{}
\keywords{Hensel's Lemma, monic polynomial, comaximal ideals,
H-prime, integral upper}
\begin{abstract} We prove a uniqueness result about the
factorization of a monic polynomial over a general
commutative ring into comaximal factors.
We apply this result to address several questions
raised by Steve McAdam. These questions, inspired by
Hensel's Lemma, concern properties of prime
ideals and the factoring of monic polynomials modulo
prime ideals.
\end{abstract}
\maketitle
\section{Introduction}
\baselineskip 16 pt
There is an interesting relationship between the factorization
of monic polynomials and the behavior of prime ideals in
integral extensions. This is illustrated for example by the
well-known result of Nagata \cite[(43.12)]{N} that asserts that
a quasilocal integral domain $R$ satisfies Hensel's Lemma if and
only if every extension domain integral over $R$ is quasilocal.
Other references that deal with this relationship include
the papers \cite{HW} and \cite{M1}.
Recent work of Steve McAdam \cite{M1}, \cite{M2}, \cite{Mabs} on
this topic is the motivation for
our interest in the matters considered here. For a prime ideal
contained in the Jacobson radical of an integral domain,
McAdam \cite{M1} introduces the concepts of H-prime, weak-H-prime
and quasi-H-prime. The H-primes are precisely those for which a
version of Hensel's Lemma holds. The other definitions reflect a
careful analysis of the comaximal factorization of monic polynomials.
In Theorem~\ref{CMF} we make use of a famous theorem of
Quillen-Suslin, a key ingredient in their resolution of
the Serre Conjecture, to prove a uniqueness
result concerning comaximal factors of a monic polynomial over a
general commutative ring. We apply this result to
prove in Theorems~\ref{WHH} and \ref{QHH} that the concepts of
H-prime, weak H-prime and quasi-H-prime are equivalent.
All rings considered here are commutative with unity.
Two general references for our notation and terminology
are \cite{N} and \cite{K}.
\section{Comaximal factors of monic polynomials}
\begin{rem}\label{QS}
Let $I,J$ be ideals in a ring $S$ for which $IJ=fS$ where $f$ is
a nonzerodivisor in $S$. Then $I,J$ are invertible ideals, i.e.,
rank-1 projective $S$-modules. Moreover, using subscript $f$ to
denote passing to the ring of fractions with respect to
the multiplicatively closed system generated by $f$, we have
$I_f = S_f = J_f$. Suppose in particular that $S=R[X]$, where
$R$ is a ring and $X$ is an indeterminate over $R$, and that
$f$ is monic in $R[X]$. Then by the Quillen--Suslin theorem
(\cite{S}; \cite{Q}; \cite[Chapter IV, Theorem~3.14]{K}),
$I,J$ are free $R[X]$-modules, i.e., principal ideals generated
by nonzerodivisors, and there are generators $p,q$ of $I,J$
respectively for which $f=pq$.
\end{rem}
\begin{thm}\label{CMF}
Let $R$ be a ring and $X$ be an indeterminate over
$R$. Let $g,h$ be comaximal monic polynomials in $R[X]$, and suppose
that the monic polynomial $f$ in $R[X]$ is such that $gh\in fR[X]$.
Then $f$ has a factorization of the form
$f=pq$ where $g\in pR[X]$
and $h\in qR[X]$. In particular, if $f$ is irreducible in $R[X]$,
then either $g\in fR[X]$ or $h\in fR[X]$.
\end{thm}
\begin{proof}
Let $R[X]_g$ and $R[X]_h$ denote the localizations of $R[X]$ at the
multiplicatively closed systems generated by $g$ and $h$ respectively,
and let $I := fR[X]_g\cap R[X]$ and $J := fR[X]_h\cap R[X]$.
Suppose $g(X)h(X)=f(X)k(X)$, where $k(X) \in R[X]$. Then because
\begin{displaymath}
h(X)=f(X)k(X)/g(X)\in I \ \ {\rm and}\ \ g(X)
=f(X)k(X)/h(X)\in J\ ,
\end{displaymath}
the ideals $I$ and $J$ are comaximal in $R[X]$; so their intersection,
which is
\begin{displaymath}
fR[X]_g\cap fR[X]_h\cap R[X] = fR[X] \ ,
\end{displaymath}
is their product. By Remark~\ref{QS},
there are generators $q(X),p(X)$ of $I,J$ respectively
for which $f=pq$, $g\in pR[X]$ and $h\in qR[X]$.
\end{proof}
\begin{rem}
With the notation of Theorem \ref{CMF}, if $R$ is
an integral domain, the polynomials $p(X), q(X) \in R[X]$
such that $f = pq$ can clearly be chosen to be
monic. More generally, if $\Spec R$ is connected,
then $p(X)$ and $q(X)$ can be chosen to be monic. To see
this, we use the fact that for any ring $S$ and nonconstant
polynomial $a(X) \in S[X]$, if $S[X]/(a)$ is a free $S$-module of
rank~$n$, then there exists a monic polynomial $b$ of
degree~$n$ in the ideal $(a)$ \cite[Prop. 2.2, page~44]{K};
and because $S[X]/(b)$ is also a free $S$-module of rank~$n$
of which $S[X]/(a)$ is a homomorphic image, $b$ is a generator of
$(a)$. Now in our present
context, we have $R[X]/(f) \cong R[X]/(p) \oplus R[X]/(q)$,
so $R[X]/(p)$ and $R[X]/(q)$ are locally free $R$-modules. Thus,
$\Spec R$ is covered by neighborhoods $\Spec R_i$ on which
the extension of $(p)$ is generated by monic polynomials.
Because $\Spec R$ is connected,
$R[X]/(p)$ has constant rank and these monic
polynomials all have the same degree $n$ and up to
units of $R_i$ are extensions
of polynomials in $(p)$ of degree $n$. An
appropriate $R$-linear combination of these polynomials in $(p)$
gives a monic polynomial of degree $n$ in
the ideal $(p)$, and this monic polynomial
generates $(p)$. Similarly, the ideal $(q)$ is generated
by a monic polynomial if $\Spec R$ is connected.
\end{rem}
\begin{exmp}
The hypotheses in Theorem~\ref{CMF} that we are working with monic
polynomials is necessary: Indeed, let $D$ be a Dedekind domain
that is not a principal ideal domain. Let $P$ be a maximal ideal
of $D$ that is not principal. Then there exists an irreducible
element $f$ in $P - P^2$. Write $fD = PQ_1^{e_1}\cdots Q_n^{e_n}$,
where the $Q_i$'s are distinct maximal ideals; because $P$ is
not principal, $n > 0$. Let
$g \in PQ_1^{e_1} \cdots Q_{n-1}^{e_{n-1}} - Q_n$, and choose
$h \in Q_n^{e_n}$ but not in any of the maximal ideals containing
$g$. Then $gD + hD = D$ and $gh \in fD$, but $g, h \not\in fD$.
For a specific example, consider the Dedekind domain $D =
{\mathbb Z}[\sqrt{-5}]$ (\cite[page 417]{A}). We have $2\cdot3
= (1+\sqrt{-5})(1-\sqrt{-5})$ in this $D$, and 2 and 3 are
comaximal, but neither of the irreducible factors on the right side
of the equation divides either 2 or 3.
\end{exmp}
\begin{exmp}
The hypothesis that $g,h$ are comaximal is very necessary:
Let $R$ be any integral domain that is not integrally closed, let
$a$ be an element of the field of fractions of
$R$ that is integral over $R$
but not in $R$, and let $f\in R[X]$ be a monic polynomial of minimal
degree of which $a$ is a root. Then $f(X)=(X-a)g(X)$ for some
polynomial $g(X)$ over the integral closure of $R$. Let $b,c$ be
nonzero elements of $R$ for which $ba\in R$ and $cg(X)\in R[X]$.
Then $f$ is irreducible in $R[X]$ and
\begin{eqnarray*}
& &f(X)(f(X)+b(X-a)+cg(X)+bc) \\
& & \qquad\qquad = (f(X)+b(X-a))(f(X)+cg(X))\ ,
\end{eqnarray*}
but $f(X)$ divides neither of the factors on the right side of the
equation.
Thus over any domain $R$ that is not integrally
closed, there exist monic polynomials $f, g, h \in R[X]$ such
that $f$ is irreducible and $gh \in fR[X]$, but
$g,h \not\in fR[X]$. For $R$ an
integrally closed domain this phenomenon is not possible; for in
this case a monic irreducible in $R[X]$ generates a prime ideal.
\end{exmp}
\section{Henselian-like conditions}
In \cite{M1}, McAdam uses the following definitions:
\begin{defn}
Let $P$ be a prime contained in the Jacobson radical of an
integral domain $R$. Then $P$ is
\begin{itemize}
\item[(a)] an \textit{H-prime} if, for every list of nonconstant
monic polynomials $f,g,h$ in $R[X]$ such that $gR[X]+hR[X]=R[X]$ and
$f-gh\in PR[X]$, there exist monic $p,q$ in $R[X]$ for which $f=pq$,
and $g-p,h-q\in PR[X]$;
\item[(b)] a \textit{weak-H-prime} if, for every list of nonconstant
monic polynomials $f,g,h$ in $R[X]$ such that $gR[X]+hR[X]=R[X]$ and
$f-gh\in PR[X]$, $f$ is reducible; and
\item[(c)] a \textit{quasi-H-prime} if, for every list of nonconstant
monic polynomials $f,g,h$ in $R[X]$ such that $gR[X]+hR[X]=R[X]$ and
$f-gh\in PR[X]$, and for every prime ideal $K$ in $R[X]$ lying over 0
in $R$ and having $f \in K$, either $K+gR[X]=R[X]$ or $K+hR[X]=R[X]$.
\end{itemize}
\end{defn}
\begin{thm}\label{WHH}
A weak-H-prime is an H-prime (and of course conversely).
\end{thm}
\begin{proof}
Let $P$ be a weak-H-prime in the domain $R$, and let $f,g,h$ be
nonconstant monic polynomials in $R[X]$ for which $f-gh\in PR[X]$
and $g,h$ generate the unit ideal in $R[X]$.
Denote by overbars images mod $PR[X]$, let
$\overline R := R/P$ and identify $R[X]/PR[X] \cong \overline R[X]$.
For each monic irreducible factor $p$ of $f$, we show that
either $\overline g$ or $\overline h$ is in $\overline p\overline R[X]$:
We have $\overline g\overline h \in \overline p \overline R[X]$
and $\overline g$ and $\overline h$ are comaximal in $\overline R[X]$.
Hence by Theorem \ref{CMF}, $\overline p$ factors into
a monic factor of $\overline g$ and a monic factor of $\overline h$.
The latter factors are comaximal because $\overline g$ and
$\overline h$ are comaximal. If both factors were nonconstant,
then because $P$ is a weak-H-prime, $p$ would be reducible; so one
of the factors is a constant, i.e., 1,
and hence either $\overline g \in \overline p\overline R[X]$
or $\overline h \in \overline p\overline R[X]$.
We proceed by induction on the number $n$ of irreducible
factors of $f$.
The case where $n = 1$ is clear.
Assume the theorem holds for monic polynomials $f$
having $n$ irreducible factors. Suppose
$f' = pf$, where $p$ is irreducible and monic, and
that $\overline{f'} = \overline{g'}\overline{h'}$,
where $g', h'$ are nonconstant, monic and comaximal.
Then by the last paragraph we may assume $\overline{g'}
\in \overline p\overline R[X]$.
Let $g$ in $R[X]$ be such that
$\overline{g'} = \overline p\overline g$, and set
$h = h'$. Then $\overline f = \overline g\overline h$
and
\begin{displaymath}
R[X] = g'R[X] + h'R[X] \subseteq gR[X] + hR[X] + PR[X]
\subseteq R[X]\ .
\end{displaymath}
Suppose there is a maximal ideal $M$ of $R[X]$ that
contains both $g$ and $h$. Then because $M$ contains
monic polynomials, it meets $R$ in a maximal ideal and
so contains $P$, a contradiction. Therefore $g, h$ are
comaximal. By the induction hypothesis, there exist
monic polynomials $g_1, h_1$ in $R[X]$ for which
$f = g_1h_1$, $\overline{g_1} = \overline g$, and
$\overline{h_1} = \overline h$; so
$f' = pf = pg_1h_1$, $\overline{pg_1}
= \overline p\overline g = \overline{g'}$ and
$\overline{h_1} = \overline h = \overline{h'}$. This
completes the induction and the proof.
\end{proof}
Much of the work needed to prove Theorem \ref{QHH} is done
by McAdam in \cite{M1}; it merely remains for us to make a
few observations and apply the Quillen-Suslin Theorem.
\begin{thm}\label{QHH}
A quasi-H-prime is an H-prime, and conversely.
\end{thm}
\begin{proof}
McAdam proves in \cite[Proposition~(2.1)]{M1} that an
H-prime is a quasi-H-prime; so it remains to prove
that a quasi-H-prime is an H-prime.
Let $P$ be a quasi-H-prime in the domain $R$, and let $f,g,h$ be
nonconstant monic polynomials in $R[X]$ such that $f-gh\in PR[X]$
and $gR[X]+hR[X] =R[X]$. Then by \cite[Proposition~(2.7)]{M1},
there are ideals $I,J$ properly containing $fR[X]$
for which
$$
\frac{I}{fR[X]} \oplus \frac{J}{fR[X]} = \frac{R[X]}{fR[X]}.
$$
It follows that $I$ and $J$ are comaximal in $R[X]$ and
intersect in $fR[X]$. Thus as in the proof of Theorem \ref{CMF},
$IJ= fR[X]$, so by Remark~\ref{QS},
$f$ is the product of generators of the principal ideals $I,J$.
Neither generator can be a unit, because $I,J$ both properly
contain $fR[X]$ and their product is $fR[X]$. Thus, $f$ is reducible;
so $P$ is a weak-H-prime and hence by Theorem~\ref{WHH} an H-prime.
\end{proof}
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\end{document}
\bibitem {} , \textit{}, \textbf{} (), --.