Math 2.0

Field theory and polynomials Theorem

Uniqueness of the complete ordered field

Builds on Complete ordered fieldOrdered field

Statement

Let FF and KK be complete ordered fields. Then there is exactly one field isomorphism φ ⁣:F→K\varphi \colon F \to K that preserves the order, that is, with φ(a)≤φ(b)\varphi(a) \le \varphi(b) whenever a≤ba \le b.

Remarks

The proof first maps the rational elements of FF to those of KK. Each other element is the supremum of the rational elements below it, and φ\varphi maps it to the supremum of their images. In fact each field isomorphism between two complete ordered fields preserves the order, because the nonnegative elements are exactly the squares.

Reading path

To understand this page from the ground up, read these 3 pages in order. The foundations come first.

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Complete ordered field, concept, no evidence levelComplete ordered fieldconceptOrdered field, definition, no evidence levelOrdered fielddefinitionUniqueness of the complete ordered field, statement, no evidence levelUniqueness of the com…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Complete ordered field concept
  2. Ordered field definition
  3. Uniqueness of the complete ordered field statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Real_number, "The complete ordered field"
About this page
Tag0G8G1TH5WK
URIm2://0G8G1TH5WK
Kindstatement, theorem
WritingW0 stub
MSC12J15
StatusThis copy of the site has no build data. Run m2 status 0G8G1TH5WK.
UsesComplete ordered field, Ordered field

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