Math 2.0

Field theory and polynomials Definition

Pythagorean field

Builds on FieldReal closureAlgebraically closed field

Definition

Let FF be a field. The field FF is Pythagorean if for all a,b∈Fa, b \in F there is c∈Fc \in F with a2+b2=c2a^2 + b^2 = c^2.

Remarks

By induction on the number of terms, every finite sum of squares in a Pythagorean field is a square. Example: ℝ\R is Pythagorean, since a2+b2≥0a^2 + b^2 \ge 0 has the real square root a2+b2\sqrt{a^2 + b^2}. More generally, every real closed field and every algebraically closed field is Pythagorean. Non-example: ℚ\Q is not Pythagorean, because 12+12=21^2 + 1^2 = 2 is not the square of a rational number. Non-example: 𝔽3\F_3 is not Pythagorean, because 12+12=21^2 + 1^2 = 2 and the squares in 𝔽3\F_3 are 00 and 11.

Reading path

To understand this page from the ground up, read these 16 pages in order. The foundations come first.

Show the path as a graph
Group, concept, no evidence levelGroupconceptReal closure, concept, no evidence levelReal closureconceptRing, definition, no evidence levelRingdefinitionCommutative ring, definition, no evidence levelCommutative ringdefinitionPolynomial ring, definition, no evidence levelPolynomial ringdefinitionUnit of a ring, definition, no evidence levelUnit of a ringdefinitionZero divisor, definition, no evidence levelZero divisordefinitionField, definition, no evidence levelFielddefinitionPolynomial evaluation, definition, no evidence levelPolynomial evaluationdefinitionIntegral domain, definition, no evidence levelIntegral domaindefinitionRoot of a polynomial, definition, no evidence levelRoot of a polynomialdefinitionIrreducible element, definition, no evidence levelIrreducible elementdefinitionSplit polynomial, definition, no evidence levelSplit polynomialdefinitionIrreducible polynomial, definition, no evidence levelIrreducible polynomialdefinitionAlgebraically closed field, definition, no evidence levelAlgebraically closed …definitionPythagorean field, definition, no evidence levelPythagorean fielddefinition
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. Real closure concept
  3. Ring definition
  4. Commutative ring definition
  5. Polynomial ring definition
  6. Unit of a ring definition
  7. Zero divisor definition
  8. Field definition
  9. Polynomial evaluation definition
  10. Integral domain definition
  11. Root of a polynomial definition
  12. Irreducible element definition
  13. Split polynomial definition
  14. Irreducible polynomial definition
  15. Algebraically closed field definition
  16. Pythagorean field definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Pythagorean_field
About this page
TagGER28JFXMR
URIm2://GER28JFXMR
Kinddefinition
WritingW0 stub
MSC12D15
DefinesPythagorean field
UsesField, Real closure, Algebraically closed field
Changed2 November 2026

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • a source with a locator (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

No page links to this page yet.