Chapter 18 - Integral Domains
Section 18.1 - Fields of Fractions
Definition. Given an integral domain \(D\), we can construct a field \(F\) containing \(D\) by stating that any \(p/q \in F\), and that any two elements \(a/b = c/d\) if and only if \(ad = bc\). We can consider this akin o a set of ordered pairs
Lemma. 18.1: The relation \((a, b) ~ (c, d) \text{ if } ad = bc\) is an equivalence relation.
Lemma. 18.2: The operations of addition and multiplication on \(F\) are well-defined.
Lemma. 18.3: The set of equivalence classes of \(S, F\) under \(~\) form a field.
Theorem. 18.4: Let \(D\) be an integral domain. Then, \(D\) can be embedded in a field of fractions \(F_D\) where any element in \(F_D\) can be expressed as the quotient of two elements in \(D\).
Additionally, \(F_D\) is unique. That is, given field \(E\) such that \(E \supset D\), there exists a map \(\psi: F_D \rightarrow D\) giving an isomorphism such that \(\psi(a) = a\) for all \(a \in D\).
Corollary. 18.6: Let \(F\) be a field of characteristic \(0\). Then, \(F\) contains a subfield isomorphic to \(\mathbb{Q}\).
Corollary. 18.6: Let \(F\) be a field of characteristic \(p\). Then, \(F\) contains a subfield isomorphic to \(\mathbb{Z}_p\).
Section 18.2 - Factorization in Integral Domains
Definition. Let \(R\) be a commutative ring with identity, and \(a, b \in R\). We say that \(a\) divides \(b\), that is, \(a | b\), if there exists some \(c \in R\) such that \(b = ac\).
Definition. A unit element is any element that has a multiplicative inverse.
Definition. Two elements \(a, b \in R\) are said to be associates if there exists some unit \(u \in R\) such that \(a = ub\).
Definition. Let \(D\) be an integral domain. A nonzero element \(p \in D\) is said to be irreducible if when given \(p = ab\), either \(a\) or \(b\) is a unit.
Definition. Let \(D\) be an integral domain. A nonzero element \(p\) is prime if when given \(p = ab\), either \(p | a\) or \(p | b\).
Definition. Given integral domain \(D\), we say that \(D\) is a Unique Factorization Domain (UFD) if it satisfies the following criteria:
- Given \(a \in D, a \neq 0\), and \(a\) is not a unit, \(a\) can be written as a product of irreducible elements in \(D\).
- Let \(a = p_1 \ldots p_r = q_1 \ldots q_s\), where \(p_i\) and \(q_i\) are all irreducible. Then, \(r = s\), and there exists some function \(\pi \in S_r\) such that \(p_i\) and \(q_{\pi(j)}\) are associates for \(j = 1, \ldots, r\).
Definition. A ring \(R\) is a principal ideal domain (PID) if every ideal of \(R\) is principal.
Lemma. 18.11: Let \(D\) be an integral domain and \(a, b \in D\). Then,
- \(a | b\) if and only if \(\langle b \rangle \subseteq \langle a \rangle\)
- \(a\) and \(b\) are associates if and only if \(\langle b \rangle = \langle a \rangle\)
- \(a\) is a unit in \(D\) if and only if \(\langle a \rangle = D\).
Theorem. 18.12: Let \(D\) be a PID, and let \(\langle p \rangle\) be a nonzero ideal in \(D\). Thus, \(\langle p \rangle\) is a maximal ideal if and only if \(p\) is irreducible.
Corollary. 18.13: Let \(D\) be a PID. For any \(p \in D\), if \(p\) is irreducible, then \(p\) is prime.
Lemma. 18.14: Let \(D\) be a PID. Let \(I_1 \subseteq I_2 \subseteq \ldots\). Then, there exists some integer \(N\) such that \(I_n = I_N\) for all \(n > N\). That is, any chain of ideals converges.
Definition. Any commutative ring that satisfies the above condition (the ascending chain condition), even if it's not a PID, is called a Noetherian ring.
Theorem. 18.15: Every PID is a UFD. Note that the converse is not true.
Corollary. 18.16: Let \(F\) be a field. Then, \(F[x]\) is a UFD.
Definition. Any integral domain \(D\) is a Euclidean domain with a Euclidean function \(nu: D \\ \{0\} \rightarrow \mathbb{N}\) that satisfies the following:
- Given \(a, b \neq 0\), then \(\nu(a) \leq \nu(ab)\).
- Given, \(a, b \in D\) and \(b \neq 0\), there exists some \(q, r \in D\) such that \(a = bq + r\) and either \(r = 0\) or \(\nu(r) < \nu(b)\).
Example. Absolute value on \(\mathbb{Z}\) is a Euclidean validation.
Example. Degree on \(F[x]\) is a Euclidean validation.
Example. \(\nu(a + bi) = a^2 + b^2\) is a Euclidean validation over \(\mathbb{Z}[i]\).
Theorem. 18.21: Every Euclidean domain is a PID.
Corollary. Every Euclidean domain is a UFD.
Definition. Given a polynomial \(p(x) \in D\), with \(D\) being an integer domain, we say that the content of \(p(x)\) is the greatest common divisor of its coefficients. Additionally, if the content is \(1\), we say that \(p(x)\) is primitive.
Theorem. 18.24: Let \(D\) be a UFD, and \(f(x), g(x) \in D[x]\) be primitive. Then, \(f(x)g(x)\) is primitive.
Lemma. 18.25: Given \(D\) is a UFD, and \(p(x), q(x) \in D[x]\), the content of \(p(x)q(x)\) is equal to the product of the contents of the individual polynomials
Lemma. 18.26: Let \(D\) be a UFD and \(F = F_D\) be its field of fractions. Given \(p(x) \in D[x]\), and \(p(x) = f(x)g(x)\) with \(f(x), g(x) \in F_D\), we can say that \(p(x) = f_1(x)g_1(x)\) with \(f_1(x), g_1(x) \in D\). Additionally, \(\deg f_1(x) = \deg f(x)\) and \(\deg g_1(x) = \deg g(x)\).
As a direct consequence, we see the following.
Corollary. Let \(D\) be a UFD, and \(F = F_D\). Then, a primitive polynomial \(p(x) \in D[x]\) is irreducible in \(D[x]\) if and only if it is irreducible in \(F[x]\).
Corollary. Let \(D\) be a UDF, and \(F = F_D\). Then, if a monic polynomial \(p(x) \ in D[x]\) can be written as \(p(x) = f(x)g(x)\) with \(f(x), g(x) \in F_D[x]\), then \(p(x)\) can be written as \(p(x) = f_1(x)g_1(x)\), where \(f_1(x), g_1(x) \in D[x]\).
Theorem. If \(D\) is as UFD, then \(D[x]\) is a UFD.
Corollary. This theorem has several corollaries:
- Given a field \(F\), since \(F\) is a PID, it is also a UFD. Thus, \(F[x]\) is a UFD.
- The ring of polynomials over integers, \(\mathbb{Z}[x]\) is a UFD.
- Given \(D\) is a UFD, \(D[x]\) is a UFD. Thus, \(D[x_1, x_2]\) is a UFD, and by induction, \(D[x_1, \ldots, x_n]\) is a UFD.