Exercise · ZW-ORDERED-FIELD-009 · ★★★ Advanced
Why can the finite field \(\mathbb Z/p\) (\(p\) prime) not be made an ordered field?
Choose one option.
Show answer
- A. Because \(1+1+\cdots+1=0\) contradicts \(1>0\)
Explanation
In an ordered field \(1>0\) and every sum \(1+\cdots+1\) is positive; in \(\mathbb Z/p\), adding \(p\) ones gives \(0\).
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-003 Which statement is true in every ordered field F?
- 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?