Integral Domains Andrew Newman December 8, 2009 16.1 – The Field of Fractions • The field of fractions of an integral domain D, FD is the set of equivalence classes on S = {(a,b): a,b є D} of ~; where (a, b) ~ (c, d) if and only if ad = bc. • Examples – Integral Domain: Z Field of Fractions: Q – Integral Domain: Z5 Field of Fractions: Z5 – Integral Domain: Z[i] Field of Fractions: Q[i] – Integral Domain: Q[x] Field of Fractions: Q(x) Mathematical Results • Lemma 16.1 – Definition of the field of fractions, and establishes ~ as an equivalence relation. • Lemma 16.2 – The operations of addition and multiplication on FD are well defined. • Lemma 16.3 – FD is a field Theorem 16.4 • Let D be an integral domain. Then D can be embedded in the field of fractions FD, where any element in FD can be expressed as the quotient of two elements in D. Furthermore, the field of fractions FD is unique in the sense that if E is any field containing D, then there exists a map ψ: FD → E giving an isomorphism with a subfield of E such that ψ(a) = a for all a є D. S E φ' D(Z) Ψ Ω -isomorphism FD(Q) φ - homomorphism Corollaries • Let F be a field of characteristic zero. Then F contains a subfield isomorphic to Q. – Show that F contains a subring isomorphic to the integers and therefore since the integers can be embedded into Q, F must contain a subfield isomorphic to Q. • Let F be a field of characteristic p. Then F contains a subfield isomorphic to Zp. 16.2 – Factorization in Integral Domains • Terms – Unit – An element in a ring that has an inverse. – Associates – Elements a and b are associates if there exists a unit u such that a = ub. – Irreducible – A nonzero element p є D that is not a unit such that whenever p = ab, either a or b is a unit. – Prime – An irreducible p such that whenever p|ab, either p|a or p|b. Examples of Irreducibles • Integral Domain: Z • Integral Domain: Z5 • Integral Domain: Q[x] Irreducibles: primes Irreducibles: none Irreducibles: irreducible polynomials Not all irreducibles are primes • Let R = {x2 * f(x,y) + xy * g(x,y) + y2 * h(x,y)} be a subring of Q[x,y]. Then x2, xy, and y2 are all irreducible however xy divides x2y2, but it doesn’t divide either x2 or y2. So xy is a nonprime irreducible. • Corollary 16.9 – Let D be a principal ideal domain, if p is irreducible, then p is prime. Types of domains • Principal ideal domain – An integral domain every ideal is a principal ideal – Example: Z – Non-example: Z[x] • Unique factorization domain – An integral domain such that any nonzero element that is not a unit can be written as the unique product of irreducible elements. • Euclidean domain – domains on which a division algorithm is defined. Principal Ideal Domains • Example: Z is a principal ideal domain because every one of its ideals (all of which are nZ, where n is a natural number) are generated by one single element. • Non-example: Z[x] is not because I = {5f(x) + xg(x)} is an ideal but it is not a principal ideal because 5 is in it and so is x, however the only element that generates both is 1 and the ideal containing 1 is the whole ring so this ideal must generate 3 which is impossible. Unique Factorization Domain • Example: Q[x] is a unique factorization domain, and in fact if F is a field then F[x] is a unique factorization domain. (Corollary 16.12 and Corollary 16.21) (proof) • Non-Example: Z[i] is not a unique factorization domain, by the counterexample 10 = 5 * 2, where 5 and 2 are irreducible and 10 = (3 + i)(3 - i) . Also (3 – i) is an example of an irreducible that is not prime as it divides 10 but not 5 or 2. Theorems about PIDs and UFDs • Lemma 16.7 Let D be an integral domain and let a, b є D. Then a |b b a . a and b are associates a is a unit in D a b a . D. • Theorem 16.8 – Let D be a PID and <p> be a nonzero ideal in D. Then <p>is a maximal ideal if and only if p is irreducible. Theorems about PIDs and UFDs • Corollary 16.9 - Let D be a PID. If p is irreducible, then p is prime. • Lemma 16.10 - Let D be a PID. Let I1, I2, … be a set of ideals such that I1 I2 … Then there exists an integer N such that In = IN for all n ≥ N. This is called the ascending chain condition (ACC). • Theorem 16.11 - Every PID is a UFD. Euclidean Domain • Let ν: D → N U {0}. This is a Euclidean valuation. A Euclidean domain has the following properties – If a and b are nonzero elements in D, then ν(a) ≤ ν(ab). – Let a,b be elements in D with b ≠ 0. Then there exists elements q and r such that a = bq + r and either r = 0 or ν(r) < ν(b). Examples of Euclidean Valuations • Absolute value (creates a Euclidean domain for the integers) • Absolute value squared for Gaussian integers. • The degree of a polynomial. Theorems • Theorem 16.13 - Every Euclidean domain is a PID. (proof) • Corollary 16.14 – Every Euclidean domain is a UFD. Factorization in D[x] • Terminology – Content – The content of a polynomial D[x] (where D is a unique factorization domain) is the greatest common divisor of a0, …, an. – Primitive – A polynomial of D[x] is a primitive if it has a content of one. • Theorem 16.15 – If f(x) and g(x) are primitive then f(x)g(x) is primitive. (proof) • Lemma 16.16 – Let D be a UFD, and let p(x) and q(x) be in D[x]. Then the content of p(x)q(x) is equal to the contents of p(x) and q(x). Theorems • Lemma 16.17 – Let D be a UFD and F, its field of fractions. Suppose that p(x) is in D[x] and p(x) = g(x)f(x) where f(x) and g(x) are in F[x]. Then p(x) = f1(x)g1(x) and f1(x) and g1(x) are in D[x]. Furthermore, deg(f(x)) = deg(f1(x)) and deg(g(x)) = deg(g1(x)). • Theorem 16.20 – If D is a UFD, then D[x] is a UFD. • Corollary 16.21 – Let F be a field. Then F[x] is a UFD. • Corollary 16.22 – Z[x] is a UFD. • Corollary 16.23 – Let D be a UFD. Then D[x1, …, xn] is a UFD.