Abstract Algebra Theory and Applications - Computer Science ...

292 CHAPTER 16 INTEGRAL DOMAINS7. Let p be prime and denote the field of fractions of Z p [x] by Z p (x). Provethat Z p (x) is an infinite field of characteristic p.8. Prove that the field of fractions of the Gaussian integers, Z[i], isQ(i) = {p + qi : p, q ∈ Q}.9. A field F is called a prime field if it has no proper subfields. If E is asubfield of F and E is a prime field, then E is a prime subfield of F .(a) Prove that every field contains a unique prime subfield.(b) If F is a field of characteristic 0, prove that the prime subfield of F isisomorphic to the field of rational numbers, Q.(c) If F is a field of characteristic p, prove that the prime subfield of F isisomorphic to Z p .10. Let Z[ √ 2 ] = {a + b √ 2 : a, b ∈ Z}.(a) Prove that Z[ √ 2 ] is an integral domain.(b) Find all of the units in Z[ √ 2 ].(c) Determine the field of fractions of Z[ √ 2 ].(d) Prove that Z[ √ 2i] is a Euclidean domain under the Euclidean valuationν(a + b √ 2 i) = a 2 + 2b 2 .11. Let D be a UFD. An element d ∈ D is a greatest common divisor of aand b in D if d | a and d | b and d is divisible by any other element dividingboth a and b.(a) If D is a PID and a and b are both nonzero elements of D, prove thereexists a unique greatest common divisor of a and b. We write gcd(a, b)for the greatest common divisor of a and b.(b) Let D be a PID and a and b be nonzero elements of D. Prove thatthere exist elements s and t in D such that gcd(a, b) = as + bt.12. Let D be an integral domain. Define a relation on D by a ∼ b if a and b areassociates in D. Prove that ∼ is an equivalence relation on D.13. Let D be a Euclidean domain with Euclidean valuation ν. If u is a unit inD, show that ν(u) = ν(1).14. Let D be a Euclidean domain with Euclidean valuation ν. If a and b areassociates in D, prove that ν(a) = ν(b).15. Show that Z[ √ 5 i] is not a unique factorization domain.16. Prove or disprove: Every subdomain of a UFD is also a UFD.