Remark 0.1. Let (F,+,·,<) be an ordered field. Building on the previous exercise,we can define the set of integers ZF in the field F viaZp = NF U{0}U{-n : n e NF},where 0 denotes the additive identity in the field.Moreover we can define an equivalence relation on the set of pairs ZFx (ZF\{0})via (a, b)define the set of rational numbers QF in the field F as the set of all equivalence~ (c, d) if and only if a · d = b. c. Then, proceeding as in lecture, we canclasses:QF ={ : ; is the equivalence class of a pair (a, b) E Zp X (ZF\ {0})}.Continuing as in lecture, we can define the operations of addition and multiplication,as well as an order relation on QF with respect to which QF is an ordered field.Let F be an ordered field with the least upper bound property. Provethat there is a unique function o : Q→ QF that satisfies the following properties:Ф(р + 9) — Ф(р) + ф(q),$(p · q) = ¢(p) · ø(g),if p < q then o(p) < ¢(q)for any p, q E Q.Hint: When constructing ø, work your way up from N, to Z, and then to Q.

An ordered field F with least upper bound property To prove that exist a unique function s.t. (i)…

Let F be an ordered field with the least upper bound property. Prove that there is a unique function o : Q → QF that satisfies the following properties: o(p q) = 0(p) · ở(q), $(p +q) = ¢(p) + ¢(q), if p < q then ø(p) < ¢(q) %3D for any p, q E Q. Hint: When constructing o, work your way up from N, to Z, and then to Q.

Let F be an ordered field with the least upper bound property. Prove that there is a unique function o : Q → QF that satisfies the following properties: o(p q) = 0(p) · ở(q), $(p +q) = ¢(p) + ¢(q), if p < q then ø(p) < ¢(q) %3D for any p, q E Q. Hint: When constructing o, work your way up from N, to Z, and then to Q.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter7: Real And Complex Numbers

Section7.1: The Field Of Real Numbers

Problem 22E: Prove that if F is an ordered field with F+ as its set of positive elements, then F+nen+, where e...

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