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(c) All real numbers of the form a+√2b, where a, b = Q, under the usual operations. (Denote this collection of reals by F.)

(c) All real numbers of the form a+√2b, where a, b = Q, under the usual operations. (Denote this collection of reals by F.)

Elements Of Modern Algebra
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 13E
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c) please.

(I think it's not associative so it cannot be a ring but I'm not sure whether that's correct). Thank you very much for the help!!

Question 1 In each of the following cases, check whether the given object is: (1) a ring, (2) a commutative ring, (3) a ring with multiplicative identity ('unity'), (4) an integral domain, and (5) a field. Show all work. (a) The collection of all complex numbers of the form r+i.q, where r, q EQ, under the usual operations. (b) Rx R, under the component-wise + and .. (c) All real numbers of the form a+√√2b, where a, b = Q, under the usual operations. (Denote this collection of reals by F.) (d) All complex numbers of the form § + in, where , n are reals in F defined in (c). (e) The set of all infinitely many times differential functions [0, 1] → R, i.e., so that f(n) exists for every n.
Transcribed Image Text:Question 1 In each of the following cases, check whether the given object is: (1) a ring, (2) a commutative ring, (3) a ring with multiplicative identity ('unity'), (4) an integral domain, and (5) a field. Show all work. (a) The collection of all complex numbers of the form r+i.q, where r, q EQ, under the usual operations. (b) Rx R, under the component-wise + and .. (c) All real numbers of the form a+√√2b, where a, b = Q, under the usual operations. (Denote this collection of reals by F.) (d) All complex numbers of the form § + in, where , n are reals in F defined in (c). (e) The set of all infinitely many times differential functions [0, 1] → R, i.e., so that f(n) exists for every n.
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