Exercise · ZW-ORDERED-FIELD-003 · ★☆☆ Basic
Which statement is true in every ordered field \(F\)?
Choose one option.
Show answer
- A. \(a^2\ge 0\) for all \(a\in F\)
Explanation
A square is never negative, but it equals \(0\) when \(a=0\); and \(a<b\) does not order squares (\(-3<2\)).
Review the concept: Ordered field
An ordered field is a field \(F\) with a total order \(<\) compatible with addition and with multiplication by positive elements.
\[a0\Rightarrow ac
More practice problems: ordered field
- ZW-ORDERED-FIELD-001 If a<b and c<0 in an ordered field, what holds?
- ZW-ORDERED-FIELD-002 Which of these sets, with the usual operations and order, is not an ordered field?
- ZW-ORDERED-FIELD-004 If a<b in an ordered field, which inequality follows?
- ZW-ORDERED-FIELD-005 Multiply the inequality -3<2 by -4. What results?
- ZW-ORDERED-FIELD-006 If 0<a<b, which is true?
- ZW-ORDERED-FIELD-007 For a,b in an ordered field, how many of a<b, a=b, a>b are true?
- ZW-ORDERED-FIELD-008 If 0<a<b in an ordered field, how do a+b and 2a compare?
- ZW-ORDERED-FIELD-009 Why can the finite field mathbb Z/p (p prime) not be made an ordered field?