Solve

10. 11. 12. 13. 14. 15. 16. not divisible by the square of a prime, prove that the norm N0: + chI) : In2 dbzl satises the four assertions made preceding Example 1. (This exercise is referred to in this chapter.) . In an integral domain, show that a and b are associates if and only if (a) : (b). . Show that the union of a chain II C I2 C - - - of ideals of a ring R is an ideal of R. (This exercise is referred to in this chapter.) . In an integral domain, show that the product of an irreducible and a unit is an irreducible. . Suppose that a and I; belong to an integral domain, I) a5 0, and a is not a unit. Show that (ab) is a proper subset of (b). (This exercise is referred to in this chapter.) . Let D be an integral domain. Dene (I ~ I; if a and b are associates. Show that this denes an equivalence relation on D. . In the notation of Example 7, show that d(xy) : d(x)d(y). . Let D be a Euclidean domain with measure at. Prove that u is a unit in B if and only if d(u) : d(1). . Let D be a Euclidean domain with measure of. Show that if a and b are associates in D, then d(a) : d(b). Let D be a principal ideal domain and let p E D. Prove that (p) is a maximal ideal in B if and only if p is irreducible. Trace through the argument given in Example 7 to nd q and r in Z[t'] such that 3 _ 4i : (2 + 55):; + r and d(r)