Fields and Ordered Fields

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last updates: 5 June 2012

Fields and Ordered Fields

A field is a set 𝔽 with operations +: 𝔽×𝔽 𝔽 (a,b) a+b and ·: 𝔽×𝔽 𝔽 (a,b) a·b=ab such that

  1. If a,b,c𝔽 then (a+b)+c =a+(b+c).
  2. If a,b𝔽 then a+b=b+a.
  3. There exists 0𝔽 such that if a𝔽 then 0+a=a+0 =a.
  4. If a𝔽 then there exists -a𝔽 such that a+(-a) =(-a) +a=0.
  5. If a,b,c𝔽 then (ab)c =a(bc).
  6. If a,b,c𝔽 then (a+b)c =ac+bc and c(a+b) =ca+cb.
  7. There exists 1𝔽 such that if a𝔽 then 1·a =a·1=a.
  8. If a𝔽 and a0 then there exists a-1𝔽 such that a·a-1 =a-1·a =1.
  9. If a,b𝔽 then ab=ba.

An ordered field is a field 𝔽 with a total order such that

  1. If a,b,c 𝔽 and ab then a+cb+c, and
  2. If a,b𝔽 and a0 and b0 then ab0,
where a<b if ab and ab, ab if ab, and a>b if ab.

The absolute value on 𝔽 is the function |n|: 𝔽 𝔽0 given by |x| =sup{x, -x}.

Let 𝔽 be an ordered field with order . Then

  1. If a𝔽 and a>0 then -a<0.
  2. If a𝔽 and a>0 then a-1>0.
  3. If a,b𝔽 and a>0 and b>0 then ab>0.
  4. If a𝔽 then a20.
  5. If a,b𝔽 and a0 and b0 then ab if and only if a2b2.
  6. 10.
  7. If x0 and y0 then x+y0.

Notes and References

These fundamental definitions and properties of ordered ordered fields are too often assumed to be true by osmosis. The basic properties of ordered fields appearing in Proposition 1.1??? are used incessantly in working with real numbers.

The definition of ordered fields is given in [Bou, Ch. VI § 2 no. 3 Definition 3]. The definition of absolute value is given in [Bou, Alg. Ch. VI § 1 no. 11 Definition 4]. These definitions also appear in the intorduction to the real numbers given in [Bou, Top. Ch. 4 § 1].


[Bou] N. Bourbaki, Algebra, Chapter 6, Masson????? MR?????.

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