Math 2.0

Field theory and polynomials Theorem

The constructible numbers form a field closed under square roots

Builds on Constructible numberSubfield

Statement

The set of the constructible numbers is a subfield of ℂ\mathbb{C}. If zz is constructible, then the square roots of zz are constructible. Moreover, the field of constructible numbers is the smallest subfield of ℂ\mathbb{C} that is closed under square roots.

Remarks

The proof of closure uses classical constructions: parallels and similar triangles give products and quotients, and a circle on a diameter of length 1+x1 + x gives x\sqrt{x}. Conversely, the intersection points of lines and circles have coordinates that come from the old ones by field operations and square roots. So a number is constructible if and only if it is obtained from ℚ\mathbb{Q} by finitely many field operations and square roots. The constructible real numbers form an ordered field in which every positive element has a square root.

Reading path

To understand this page from the ground up, read these 10 pages in order. The foundations come first.

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Group, concept, no evidence levelGroupconceptStraightedge and compass construction, concept, no evidence levelStraightedge and comp…conceptRing, definition, no evidence levelRingdefinitionConstructible number, definition, no evidence levelConstructible numberdefinitionCommutative ring, definition, no evidence levelCommutative ringdefinitionSubring, definition, no evidence levelSubringdefinitionUnit of a ring, definition, no evidence levelUnit of a ringdefinitionField, definition, no evidence levelFielddefinitionSubfield, definition, no evidence levelSubfielddefinitionThe constructible numbers form a field closed under square roots, statement, no evidence levelThe constructible num…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.
  • no evidence level
  1. Group concept
  2. Straightedge and compass construction concept
  3. Ring definition
  4. Constructible number definition
  5. Commutative ring definition
  6. Subring definition
  7. Unit of a ring definition
  8. Field definition
  9. Subfield definition
  10. The constructible numbers form a field closed under square roots statement

Where this leads

6 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Constructible_number, Algebraic properties
  2. Web https://en.wikipedia.org/wiki/Constructible_number, Equivalence of algebraic and geometric definitions
About this page
Tag0PN9WMRWT7
URIm2://0PN9WMRWT7
Kindstatement, theorem
WritingW0 stub
MSC12F05
StatusThis copy of the site has no build data. Run m2 status 0PN9WMRWT7.
UsesConstructible number, Subfield

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