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(a) The set of rational numbers Q can be embedded in every ordered integral domain. (b) A finite integral domain cannot be ordered. (c) The relation defined by (k, 1)~ (m, n), V(k, 1), (m, n) D x D# in an integral domain D is reflexive for the following reason: (k, 1)~ (k, 1) ⇒ k·l=l·k⇒l- k = k·l⇒ (l, k) ~ (l, k). (d) The unity e € D is the least positive element of every ordered integral domain D. (e) The field of rational numbers contains a subfield isomorphic to .

(a) The set of rational numbers Q can be embedded in every ordered integral domain. (b) A finite integral domain cannot be ordered. (c) The relation defined by (k, 1)~ (m, n), V(k, 1), (m, n) D x D# in an integral domain D is reflexive for the following reason: (k, 1)~ (k, 1) ⇒ k·l=l·k⇒l- k = k·l⇒ (l, k) ~ (l, k). (d) The unity e € D is the least positive element of every ordered integral domain D. (e) The field of rational numbers contains a subfield isomorphic to .

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. (a) The set of rational numbers Q can be embedded in every ordered integral domain. (b) A finite integral domain cannot be ordered. (c) The relation defined by (k, 1) ~ (m, n), V(k, 1), (m, n) € D × D# in an integral domain D is reflexive for the following reason: (k, 1)~ (k, l) → k·l=l-k⇒l·k = k·l⇒ (l, k) ~ (l, k). (d) The unity e € D is the least positive element of every ordered integral domain D. (e) The field of rational numbers <Q, +, > contains a subfield isomorphic to <Z, +, . >.
Transcribed Image Text:Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. (a) The set of rational numbers Q can be embedded in every ordered integral domain. (b) A finite integral domain cannot be ordered. (c) The relation defined by (k, 1) ~ (m, n), V(k, 1), (m, n) € D × D# in an integral domain D is reflexive for the following reason: (k, 1)~ (k, l) → k·l=l-k⇒l·k = k·l⇒ (l, k) ~ (l, k). (d) The unity e € D is the least positive element of every ordered integral domain D. (e) The field of rational numbers <Q, +, > contains a subfield isomorphic to <Z, +, . >.
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All the exercises have been answered. Part (c) is an incorrect question as the relation ~ is not defined properly.

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