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Question 2. S:=, and a field F:= F, +,.>. Given an integral domain D:=, a ring (a) Prove that the integral domain D is a field if and only if each equation ax = = b for a,b € D and a 0 has a unique solution in D. (b) Given a ring S. For each s ES define a mapping , : S→S by o, (a) = sa for each a € S. If S is a commutative ring, explain why Kero, # {0} if and only if s= 0 or s is a zero divisor in S. (c) For S a field and s0, explain why , defined in (b) is an isomorphism of the additive group of S onto itself.

Question 2. S:=, and a field F:= F, +,.>. Given an integral domain D:=, a ring (a) Prove that the integral domain D is a field if and only if each equation ax = = b for a,b € D and a 0 has a unique solution in D. (b) Given a ring S. For each s ES define a mapping , : S→S by o, (a) = sa for each a € S. If S is a commutative ring, explain why Kero, # {0} if and only if s= 0 or s is a zero divisor in S. (c) For S a field and s0, explain why , defined in (b) is an isomorphism of the additive group of S onto itself.

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter8: Polynomials
Section8.6: Algebraic Extensions Of A Field
Problem 12E
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Question 2. SS, +, >, and a field F:= F, +,.>. Given an integral domain D:= D, +, >, a ring (a) Prove that the integral domain D is a field if and only if each equation ax = = b for a, b € D and a 0 has a unique solution in D. (b) Given a ring S. For each s € S define a mapping ,: S→S by a, (a) = sa for each a € S. If S is a commutative ring, explain why Kero, {0} if and only if s= 0 or s is a zero divisor in S. (c) For S a field and s 0, explain why o, defined in (b) is an isomorphism of the additive group of S onto itself. (d) Prove that if C denotes any collection of subfields of a field F, then the intersection of all the fields in C is also a subfield of F.
Transcribed Image Text:Question 2. SS, +, >, and a field F:= F, +,.>. Given an integral domain D:= D, +, >, a ring (a) Prove that the integral domain D is a field if and only if each equation ax = = b for a, b € D and a 0 has a unique solution in D. (b) Given a ring S. For each s € S define a mapping ,: S→S by a, (a) = sa for each a € S. If S is a commutative ring, explain why Kero, {0} if and only if s= 0 or s is a zero divisor in S. (c) For S a field and s 0, explain why o, defined in (b) is an isomorphism of the additive group of S onto itself. (d) Prove that if C denotes any collection of subfields of a field F, then the intersection of all the fields in C is also a subfield of F.
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