Question #14972

Give an example of incomplete field that may be ordered.

Expert's answer

Every ordered field is a formally real field, i.e., 0 cannot be written as a sum of nonzero squares.

Conversely, every formally real field can be equipped with a compatible total order, that will turn it into an ordered field. (This order is often not uniquely determined.)

Finite fields and more generally fields of finite characteristic cannot be turned into ordered fields, because in characteristic p, the element -1 can be written as a sum of (p-1) squares 1². The complex numbers also cannot be turned into an ordered field, as −1 is a square (of the imaginary number i) and would thus be positive. Also, the p-adic numbers cannot be ordered, since Q2 contains a square root of −7 and Qp (p > 2) contains a square root of 1 − p.

Conversely, every formally real field can be equipped with a compatible total order, that will turn it into an ordered field. (This order is often not uniquely determined.)

Finite fields and more generally fields of finite characteristic cannot be turned into ordered fields, because in characteristic p, the element -1 can be written as a sum of (p-1) squares 1². The complex numbers also cannot be turned into an ordered field, as −1 is a square (of the imaginary number i) and would thus be positive. Also, the p-adic numbers cannot be ordered, since Q2 contains a square root of −7 and Qp (p > 2) contains a square root of 1 − p.

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