Ring Theory 6: Introduction to Fields
Definition of a field and examples. Practice problems: 1) Prove that a finite integral domain is a field. 2) Let d be a fixed integer. Prove that the set {a + b*sqrt(d) | a,b are rationals} is a field (under normal addition and multiplication). This field is referred to as Q(sqrt(d)). Partial solutions to problems from video 5: 1) Units of Z_20: {1,3,7,9,11,13,17,19}. Zero Divisors of Z_20: {2,4,5,6,8,10,12,14,15,16,18}. The relationship is hinted at in problem 2 of video 5. 3) We already proved this set is a ring in a previous practice problem. Just show it has a unity, it is commutative, and has no zero divisors. 4) ab = ac implies ab - ac = 0 implies a(b - c) = 0 It is assumed this is an integral domain, so either a=0 or b - c = 0 But we assumed a is nonzero, so it must be that b - c = 0, and so b = c
Definition of a field and examples. Practice problems: 1) Prove that a finite integral domain is a field. 2) Let d be a fixed integer. Prove that the set {a + b*sqrt(d) | a,b are rationals} is a field (under normal addition and multiplication). This field is referred to as Q(sqrt(d)). Partial solutions to problems from video 5: 1) Units of Z_20: {1,3,7,9,11,13,17,19}. Zero Divisors of Z_20: {2,4,5,6,8,10,12,14,15,16,18}. The relationship is hinted at in problem 2 of video 5. 3) We already proved this set is a ring in a previous practice problem. Just show it has a unity, it is commutative, and has no zero divisors. 4) ab = ac implies ab - ac = 0 implies a(b - c) = 0 It is assumed this is an integral domain, so either a=0 or b - c = 0 But we assumed a is nonzero, so it must be that b - c = 0, and so b = c