64 662
Assignments Done
99,2%
Successfully Done
In September 2018

Answer to Question #14972 in Calculus for Paul

Question #14972
Give an example of incomplete field that may be ordered.
Expert's answer
Every ordered field is a formally real field, i.e., 0 cannot be written as a sum of nonzero squares.
Conversely, every formally real field can be equipped with a compatible total order, that will turn it into an ordered field. (This order is often not uniquely determined.)
Finite fields and more generally fields of finite characteristic cannot be turned into ordered fields, because in characteristic p, the element -1 can be written as a sum of (p-1) squares 1². The complex numbers also cannot be turned into an ordered field, as −1 is a square (of the imaginary number i) and would thus be positive. Also, the p-adic numbers cannot be ordered, since Q2 contains a square root of −7 and Qp (p > 2) contains a square root of 1 − p.



Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be first!

Leave a comment

Ask Your question

Submit
Privacy policy Terms and Conditions