Question 5. (a) Given the mapping 0 : C→C defined by 0(a + bi) = a -bi for all a, b € R. Show that is an automorphism of C. (b) Express the polynomial 3x4 - 4x³+4x -3 as a product of factors over R. Then express this polynomial as a product of factors over C. (c) Prove or disprove: The integral domain Z[√5], +, > is well-ordered.

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Question 5.
(a) Given the mapping 0 : C → C defined by 0(a + bi) = a - bi for all a,b € R. Show that
is an automorphism of C.
(b) Express the polynomial 3x¹ - 4r³ + 4r-3 as a product of factors over R. Then express
this polynomial as a product of factors over C.
(c) Prove or disprove: The integral domain Z[√5], +, > is well-ordered.
(d) Suppose that D = D, +, > is an ordered integral domain. Let D denote the set of all
onzero elements not in D. Prove that if D is well ordered, then every nonempty subset
SCD has a greatest element.
Transcribed Image Text:Question 5. (a) Given the mapping 0 : C → C defined by 0(a + bi) = a - bi for all a,b € R. Show that is an automorphism of C. (b) Express the polynomial 3x¹ - 4r³ + 4r-3 as a product of factors over R. Then express this polynomial as a product of factors over C. (c) Prove or disprove: The integral domain Z[√5], +, > is well-ordered. (d) Suppose that D = D, +, > is an ordered integral domain. Let D denote the set of all onzero elements not in D. Prove that if D is well ordered, then every nonempty subset SCD has a greatest element.
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