Math 2.0

Number theory Theorem

The algebraic numbers form a field

Builds on Algebraic elementTower lawFinite extensions are algebraicSubfield

Statement

Let ℚ‾\overline{\mathbb{Q}} be the set of the complex numbers that are algebraic over ℚ\mathbb{Q}. Then ℚ‾\overline{\mathbb{Q}} is a subfield of ℂ\mathbb{C}.

Remarks

The proof: if a,ba, b are algebraic over ℚ\mathbb{Q}, then ℚ(a,b)\mathbb{Q}(a, b) is a finite extension of ℚ\mathbb{Q}, by the tower law. So a+ba + b, a−ba - b, aba b and a/ba/b for b≠0b \neq 0 are algebraic, because finite extensions are algebraic. More generally, for every extension L/KL/K, the elements of LL that are algebraic over KK form a subfield of LL. The field ℚ‾\overline{\mathbb{Q}} is countable and algebraically closed. It is an algebraic closure of ℚ\mathbb{Q}.

Reading path

To understand this page from the ground up, read these 19 pages in order. The foundations come first.

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Group, concept, no evidence levelGroupconceptRing, definition, no evidence levelRingdefinitionCommutative ring, definition, no evidence levelCommutative ringdefinitionPolynomial ring, definition, no evidence levelPolynomial ringdefinitionSubring, definition, no evidence levelSubringdefinitionUnit of a ring, definition, no evidence levelUnit of a ringdefinitionPolynomial evaluation, definition, no evidence levelPolynomial evaluationdefinitionField, definition, no evidence levelFielddefinitionRoot of a polynomial, definition, no evidence levelRoot of a polynomialdefinitionSubfield, definition, no evidence levelSubfielddefinitionField extension, definition, no evidence levelField extensiondefinitionDegree of a field extension, definition, no evidence levelDegree of a field ext…definitionAlgebraic element, definition, no evidence levelAlgebraic elementdefinitionIntermediate field, definition, no evidence levelIntermediate fielddefinitionFinite extension, definition, no evidence levelFinite extensiondefinitionAlgebraic extension, definition, no evidence levelAlgebraic extensiondefinitionTower law, statement, no evidence levelTower lawstatementFinite extensions are algebraic, statement, no evidence levelFinite extensions are…statementThe algebraic numbers form a field, statement, no evidence levelThe algebraic numbers…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. Ring definition
  3. Commutative ring definition
  4. Polynomial ring definition
  5. Subring definition
  6. Unit of a ring definition
  7. Polynomial evaluation definition
  8. Field definition
  9. Root of a polynomial definition
  10. Subfield definition
  11. Field extension definition
  12. Degree of a field extension definition
  13. Algebraic element definition
  14. Intermediate field definition
  15. Finite extension definition
  16. Algebraic extension definition
  17. Tower law statement
  18. Finite extensions are algebraic statement
  19. The algebraic numbers form a field statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Algebraic_number, Field
  2. Web https://en.wikipedia.org/wiki/Algebraic_element, Properties
About this page
Tag0QFTETCHKB
URIm2://0QFTETCHKB
Kindstatement, theorem
WritingW0 stub
MSC11R04
StatusThis copy of the site has no build data. Run m2 status 0QFTETCHKB.
UsesAlgebraic element, Tower law, Finite extensions are algebraic, Subfield

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