Section 1.8 Principal Ideal Domains and Euclidean Domains
Guiding Questions.
In this section, we’ll seek to answer the questions:
In
Section 1.5, we began a structural exploration of the notion of ring by considering a particularly important class of subrings, known as
ideals. These turn out to be integral to our understanding of factorization. Of particular importance to our understanding of factorization are
principal ideals, and integral domains in which every ideal is principal.
Definition 1.8.1.
An integral domain \(R\) in which every ideal is principal is known as a principal ideal domain (PID).
Theorem 1.8.2.
The ring \(\Z\) is a principal ideal domain.
Hint.
Use properties specific to
\(\Z\text{,}\) perhaps from
Section A.
Activity 1.8.3.
Find an integer \(d\) such that \(I = \ideal{d}\subseteq \Z\text{,}\) if
\(\displaystyle I = \setof{4x+10y}{x,y\in\Z}\)
\(\displaystyle I = \setof{6s+7t}{s,t\in\Z}\)
\(\displaystyle I = \setof{9w+12z}{w,z\in\Z}\)
\(\displaystyle I = \setof{am+bn}{m,n\in\Z}\)
You do not need to prove that each of the sets above are ideals (though you should make sure you can do it).
Challenge 1.8.4.
The ring \(\Z[x]\) is not a PID.
Hint.
Consider the ideal \(I = \langle 2, x \rangle\text{.}\)
Theorem 1.8.5.
Let \(R\) be a principal ideal domain and \(x,y\in R\) be not both zero. Let \(I = \setof{xm+yn}{m,n\in R}\text{.}\) Then:
\(I\) is an ideal, so there is some \(d\in R\) for which \(I = \ideal{d}\text{;}\)
We conclude that there exist \(s,t\in R\) such that \(d = xs + yt\text{.}\)
We have so far abstracted and axiomatized several important algebraic properties of
\(\Z\) that we discussed in
§ A. In particular, we have our usual operations of addition and multiplication, and their interactions; we have notions of divisibility/factorization, irreducibility, and primality; we also have cancellation and greatest common divisors.
Our last major abstraction from \(\Z\) is the division algorithm. The main obstacle to postulating domains with a division algorithm is a clear notion of comparison relations. That is, if \(R\) is an arbitrary domain with \(r,s\in R\text{,}\) is it possible to clearly and sensibly say which of \(r\) or \(s\) is “bigger” ? (Recall that this was a requirement for the division algorithm with nonzero remainders.) However, if there is a way to relate elements of a domain \(R\) to \(\N_0\text{,}\) we can sensibly define a division algorithm.
Definition 1.8.6.
Let
\(R\) be an integral domain. We call
\(R\) a
Euclidean domain if there is a function
\(\delta : R\setminus \set{0} \to \N_0\) such that:
If \(a,b\in R\setminus \set{0}\text{,}\) then \(\delta(a) \le \delta(ab)\text{.}\)
If \(a,b\in R\text{,}\) \(b\ne 0\text{,}\) then there exist \(q,r\in R\) such that \(a = bq+r\text{,}\) where either \(r = 0\) or \(\delta(r) \lt \delta(b)\text{.}\)
We call the function \(\delta\) a norm for \(R\text{.}\)
Thus, a Euclidean domain is an integral domain with a division algorithm that behaves in a familiar way. In the remainder of this section, we will investigate the properties of Euclidean domains. First, we consider some examples.
Theorem 1.8.7.
The field \(\Q\) is a Euclidean domain under ordinary addition and multiplication, with \(\delta(x) = 0\) for all \(x\in \Q\text{.}\)
Investigation 1.8.8.
Is \(\Z\) a Euclidean domain? If so, what is the norm function \(\delta\text{,}\) and why does this function have the required properties of a norm?
Lemma 1.8.9.
Let \(F\) be a field and \(S\subseteq F[x]\) a set containing a nonzero polynomial. Prove that \(S\) contains a polynomial \(f\) such that \(\deg(f) \le \deg(g)\) for all nonzero \(g\in S\text{.}\)
Lemma 1.8.10.
Let \(F\) be a field and \(f(x),g(x)\in F[x]\) with \(g(x)\ne 0\text{.}\) If \(\deg f(x) \ge \deg g(x) > 0\text{,}\) and \(f(x) = a_0 + a_1 x + \cdots + a_m x^m\) and \(g(x) = b_0 + b_1 x + \cdots + b_n x^n\text{,}\) then \(h(x) = f(x) - a_m b_n^{-1} x^{m-n} g(x)\) has degree strictly less than \(\deg f(x)\text{.}\)
Theorem 1.8.11. (Polynomial Division Algorithm).
Let \(F\) be a field and \(f(x),g(x)\in F[x]\) with \(g(x)\ne 0\text{.}\) Then there exist unique \(q(x),
r(x) \in F[x]\) such that
\begin{equation*}
f(x) = g(x) q(x) + r(x)\text{,}
\end{equation*}
where \(\deg(r(x)) \lt \deg g(x)\text{.}\)
Hint.
For existence, consider three cases:
\(f(x) = 0\text{;}\) \(f(x) \ne 0\) and
\(\deg f \lt \deg g\text{;}\) \(f(x) \ne 0\) and
\(\deg f \ge \deg g\text{.}\) In the last case, use induction on
\(m = \deg f(x)\text{.}\) For uniqueness, mimic the uniqueness proof of
Theorem 1.2.8.
Theorem 1.8.12.
Let \(F\) be a field. Then the ring \(F[x]\) is a principal ideal domain.
Hint.
Investigation 1.8.13.
Is \(F[x]\) a Euclidean domain for all fields \(F\text{?}\) If so, what is the norm function \(\delta\text{,}\) and why does this function have the required properties of a norm? If not, why not? Prove your answer.
In fact, every Euclidean domain is a PID.
Theorem 1.8.14.
Every Euclidean domain is a principal ideal domain.
Hint.
Exploration 1.8.15.
Where do Euclidean domains and PIDs fit in the hierarchy of abstraction found in
Question 1.7.14?
