(2a) Prove that product and sum are well-defined, i.e., if f/g = f₁/9₁ and h/k = h₁/k₁, then f/g* h/k = f1/91 * h₁/k₁ for * = - and * = +. (2b) Prove that C(x) is a field.
(2a) Prove that product and sum are well-defined, i.e., if f/g = f₁/9₁ and h/k = h₁/k₁, then f/g* h/k = f1/91 * h₁/k₁ for * = - and * = +. (2b) Prove that C(x) is a field.
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter5: Rings, Integral Domains, And Fields
Section5.3: The Field Of Quotients Of An Integral Domain
Problem 10E: Since this section presents a method for constructing a field of quotients for an arbitrary integral...
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The question is in the attached image, please if able give explanation with the taken steps, Im new to abstract algebra.
![One may construct the rational numbers Q from the integers Z as equivalence classes of pairs of integers
(a, b) with b 0 for the equivalence relation (a, b)~ (a₁, b₁) ⇒ ab₁ = a₁b. Equivalence classes
are written as a/b and called 'fractions.
Write C for the field of complex numbers and C[x] for the polynomial ring in the variable x over C.
As above, we consider
C(x) = {f/g: f, g € C[x], g = 0}
where 'f/g' is a new symbol for the equivalence class of a pair of polynomials (f, g) with g 0 for the
equivalence relation
(f,g) ~ (f1,91)
f/g = f1/91
Define product and sum by
fg1 = fig.
f/g h/k (fh)/(gk) and f/g+h/k := (fk+gh)/(gk).
fi/91 and h/k
(2a) Prove that product and sum are well-defined, i.e., if f/g
f/g* h/k = f1/91 * h₁/k₁ for * = and * = +.
(2b) Prove that C(x) is a field.
=
=
h₁/k₁, then](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8e237d3f-b8e6-4775-a6f9-5671b153aef2%2F3c4fb5fd-4d75-460a-baed-7ba92b3bcf3c%2Fg0f7z1_processed.png&w=3840&q=75)
Transcribed Image Text:One may construct the rational numbers Q from the integers Z as equivalence classes of pairs of integers
(a, b) with b 0 for the equivalence relation (a, b)~ (a₁, b₁) ⇒ ab₁ = a₁b. Equivalence classes
are written as a/b and called 'fractions.
Write C for the field of complex numbers and C[x] for the polynomial ring in the variable x over C.
As above, we consider
C(x) = {f/g: f, g € C[x], g = 0}
where 'f/g' is a new symbol for the equivalence class of a pair of polynomials (f, g) with g 0 for the
equivalence relation
(f,g) ~ (f1,91)
f/g = f1/91
Define product and sum by
fg1 = fig.
f/g h/k (fh)/(gk) and f/g+h/k := (fk+gh)/(gk).
fi/91 and h/k
(2a) Prove that product and sum are well-defined, i.e., if f/g
f/g* h/k = f1/91 * h₁/k₁ for * = and * = +.
(2b) Prove that C(x) is a field.
=
=
h₁/k₁, then
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