Could you please solve this problem for me? thank you!! Prove that in any ordered field F, a? + 1 > 0 for all a e F. Conclude that any fieldwith a solution to the equation x? + 1 = 0 in the field cannot be ordered.2.

Let F be any ordered field. We need to prove that for any a ∈ F, a2+1 > 0. We know that in an…

Answered: 2. Prove that in any ordered field F,…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Related questions

Question

Could you please solve this problem for me? thank you!!

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 $a \in F, a^{2} + 1 > 0$ .

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

Also, we know that for every nonzero element $a$ of an ordered field, we have $a^{2} > 0$ .

To prove $a^{2} + 1 > 0$ , we will consider two cases.

Case I)

If $a = 0$

$\Rightarrow a^{2} + 1 = 0^{2} + 1 = 0 + 1 = 1$ .

$\Rightarrow a^{2} + 1 = 1 > 0$

Thus, the result is true when $a = 0$ .

Trending now

This is a popular solution!

Step by step

Solved in 3 steps

SEE SOLUTION Check out a sample Q&A here

Similar questions

Recommended textbooks for you

Advanced Engineering Mathematics

Advanced Math

ISBN:

9780470458365

Author:

Erwin Kreyszig

Publisher:

Wiley, John & Sons, Incorporated

Numerical Methods for Engineers

Advanced Math

ISBN:

9780073397924

Author:

Steven C. Chapra Dr., Raymond P. Canale

Publisher:

McGraw-Hill Education

Introductory Mathematics for Engineering Applicat…

Advanced Math

ISBN:

9781118141809

Author:

Nathan Klingbeil

Publisher:

WILEY

Advanced Engineering Mathematics

Advanced Math

ISBN:

9780470458365

Author:

Erwin Kreyszig

Publisher:

Wiley, John & Sons, Incorporated

Numerical Methods for Engineers

Advanced Math

ISBN:

9780073397924

Author:

Steven C. Chapra Dr., Raymond P. Canale

Publisher:

McGraw-Hill Education

Introductory Mathematics for Engineering Applicat…

Advanced Math

ISBN:

9781118141809

Author:

Nathan Klingbeil

Publisher:

WILEY

Mathematics For Machine Technology

Advanced Math

ISBN:

9781337798310

Author:

Peterson, John.

Publisher:

Cengage Learning,

Basic Technical Mathematics

Advanced Math

ISBN:

9780134437705

Author:

Washington

Publisher:

PEARSON

Topology

Advanced Math

ISBN:

9780134689517

Author:

Munkres, James R.

Publisher:

Pearson,

SEE MORE TEXTBOOKS

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.