perplexus.info :: Just Math : Non ordered field

Non ordered field (Posted on 2007-06-06)

Prove that the field of complex numbers cannot be ordered.

An ordered field F is a field having a subset P satisfying the following:

1) For all x,y in P, x+y in P
2) For all x,y in P, x*y in P
3) For all x in F, exactly one of the statements

      (x in P,  x = 0,  -x in P) is true.

Submitted by Bractals

Rating: 3.0000 (1 votes)

Solution: (Hide)

Clearly i != 0. By the third constraint i or -i must be in P.

By the second constraint -1 = i*i = (-i)*(-i) must be in P.

By the second constraint 1 = (-1)*(-1) must be in P.

By the first constraint 0 = (1)+(-1) must be in P.

But, this is a contradiction since 0 = 0 and 0 in P cannot be both true.

Therefore, the field of complex numbers cannot be ordered.

Comments: ( You must be logged in to post comments.)

Subject	Author	Date
Solution	K Sengupta	2007-06-07 04:57:56
Solution	Robby Goetschalckx	2007-06-06 09:57:32

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