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Ordered fields are fields which have an additional structure, a linear order compatible with the field structure. This tag is for questions regarding ordered fields and their properties, as well proofs related to un-orderability of certain fields.

Let $F$ be a field, then $F$ is an ordered field if there is a binary relation $\leq$ on $F$ with the following properties:

  1. $\leq$ is a linear (total) order $F$.
  2. If $x\leq y$ then $x+z\leq y+z$.
  3. If $0\leq x,y$ then $0\leq ab$.

For example, $\Bbb Q$ and $\Bbb R$ are ordered fields.

Not every field can be ordered. For example, one can verify that if $F$ is an ordered field then the characteristics of $F$ is zero.

If $F$ is an ordered field then all the squares are positive, and so are their sums. Since $-1$ is not positive we have that in an ordered field, $-1$ is not the sum of squares, and therefore algebraically closed fields (such as $\Bbb C$) are not ordered fields.