5.4 5.4 J.A.Beachy 1 Quotient Fields from A Study Guide for Beginner’s by J.A.Beachy, a supplement to Abstract Algebra by Beachy / Blair 15. Let F be a field. Explain why Q(F ) is isomorphic to F . Why can’t we just say that Q(F ) = F ? Solution: Formally, Q(F ) is a set of equivalence classes of ordered pairs of elements of F , so it is not simply equal to the original set F . In the general construction, we identified d ∈ D with the equivalence class [d, 1], and used this to show that D is isomorphic to a subring of Q(D). When D is a field, the equivalence class [a, b] is equal to the equivalence class [ab−1 , 1] since a · 1 = b · ab−1 . This shows that when D is a field we have Q(D) ∼ = D. 16. Find the quotient field of Z2 [x]. Solution: We need to formally invert all nonzero polynomials in Z2 [x]. This produces what look like rational functions, though we do not think of them as functions taking values in Z2 since as functions things would collapse. (Remember that as functions f (x) we have x2 = x.) The set of fractions of the form such that f (x), g(x) ∈ Z2 [x], g(x) where g(x) is not the zero polynomial, is denoted by Z2 (x), and is called the rational function field over Z2 . 17. Prove that if D1 and D2 are isomorphic integral domains, then Q(D1 ) ∼ = Q(D2 ). Solution: Let D1 and D2 be isomorphic integral domains, with isomorphism α : D1 → D2 and β = α−1 . Let φ1 : D1 → Q(D1 ) and φ2 : D2 → Q(D2 ) be the standard mappings defined in Theorem 5.4.6. Recall that we have the following diagram from Theorem 5.4.6, in which any one-toone ring homomorphism θ : D → F , where F is a field, gives rise to a unique ring b = θ. homomorphism θb : Q(D) → F such that θφ D φ- Q(D) @ θ @ R @ θb ? F In the above diagram, we first let D = D1 , φ = φ1 , F = Q(D2 ), and θ = φ2 α. We d therefore get φd 2 α : Q(D1 ) → Q(D2 ) with φ2 αφ1 = φ2 α. Similarly, letting D = D2 , d φ = φ2 , F = Q(D1 ), and θ = φ1 β we get φd 1 β : Q(D2 ) → Q(D1 ) with φ1 βφ2 = φ1 β. In following this part of the argument, it may help to draw the corresponding diagrams. d We claim that φd 2 α is the isomorphism we are looking for, with inverse φ1 β. To show this, in the original diagram take D = D1 , φ = φ1 , F = Q(D1 ), and θ = φ1 . Then the 5.4 J.A.Beachy 2 c1 , since 1Q(D ) φ1 = φ1 , and it identity mapping from Q(D1 ) to Q(D1 fits the role of φ 1 is the unique ring homomorphism for which this is true. d dd On the other hand, consider φd 1 β φ2 α : Q(D1 ) → Q(D2 ). We have (φ1 β φ2 α)φ1 = φd 1 β(φ2 α) = (φ1 β)α = φ1 (βα) = φ1 , and so the uniqueness guaranteed by Theod rem 5.4.6 implies that φd 1 β φ2 α is the identity mapping. A similar argument shows d that φd 2 αφ1 β : Q(D2 ) → Q(D1 ) is the identity mapping. This completes the proof that Q(D1 ) ∼ = Q(D2 ). Comment: It is also possible to directly define an isomorphism θ : Q(D1 ) → Q(D2 ) by setting θ([a, b]) = [α(a), α(b)]. The argument given above is a standard one for any “universal” construction that guarantees the existence of a unique mapping. If you continue your study of mathematics, you are likely to encounter arguments like this in more general situations. ANSWERS AND HINTS √ 3i]. 18. Find the quotient field of Z[ √ Answer: Q( 3i) m n 19. Find the quotient field of D = m, n ∈ Z . −3n m a b a, b ∈ Q . To apply Corollary 5.4.7, show that every Hint: Let F = −3b a −1 m n d 0 element of F can be written in the form for some m, n, d ∈ Z. −3n m 0 d Answer: Q(D) ∼ = F. m n 20. Find the quotient field of D = m, n ∈ Z . −n m a b a, b ∈ Q Answer: Q(D) = −b a