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2. Prove that in any ordered field F, a? + 1 > 0 for all a e F. Conclude that any field with a solution to the equation x2 + 1 = 0 in the field cannot be ordered.

2. Prove that in any ordered field F, a? + 1 > 0 for all a e F. Conclude that any field with a solution to the equation x2 + 1 = 0 in the field cannot be ordered.

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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Prove that in any ordered field F, a? + 1 > 0 for all a e F. Conclude that any field with a solution to the equation x? + 1 = 0 in the field cannot be ordered. 2.
Transcribed Image Text:Prove that in any ordered field F, a? + 1 > 0 for all a e F. Conclude that any field with a solution to the equation x? + 1 = 0 in the field cannot be ordered. 2.
Expert Solution
Step 1

Let F be any ordered field.

We need to prove that for any a ∈ F,   a2+1 > 0.

We know that in an ordered field, 1 > 0.

Also, we know that for every nonzero element a of an ordered field, we have  a2 > 0.

To prove a2 + 1 > 0, we will consider two cases.

Case I) 

If a = 0

⇒ a2 + 1 = 02 + 1 = 0 + 1 = 1.

⇒a2 + 1 = 1 > 0

Thus, the result is true when a = 0.

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