Math 2.0

statement theorem

Existence of an algebraic closure

Statement

Every field KK has an algebraic closure.

Remarks

One proof: take a polynomial ring RR over KK with one variable for each root of each monic irreducible polynomial over KK. A maximal ideal of RR that contains the relations gives a field K1K_1 in which every polynomial over KK splits. Repeating the step gives fields K⊆K1⊆K2⊆…K \subseteq K_1 \subseteq K_2 \subseteq \dots, and their union is an algebraic closure of KK. The proof uses Zorn's lemma, through the existence of maximal ideals. The weaker ultrafilter lemma is enough. For K=QK = \mathbb{Q} or K=RK = \mathbb{R}, the complex numbers give a direct proof.

Explanations

L1 human

Every field sits inside an algebraically closed field that is algebraic over it. The general proof needs a form of the axiom of choice, because a field can have very many polynomials.

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What this needs

To read this node, read these 30 nodes in this order. The foundations come first, and the list ends with this node.

Group, concept, no evidence levelGroupconceptRing, definition, no evidence levelRingdefinitionCommutative ring, definition, no evidence levelCommutative ringdefinitionIdeal, definition, no evidence levelIdealdefinitionPolynomial ring, definition, no evidence levelPolynomial ringdefinitionSubring, definition, no evidence levelSubringdefinitionUnit of a ring, definition, no evidence levelUnit of a ringdefinitionZero divisor, definition, no evidence levelZero divisordefinitionMaximal ideal, definition, no evidence levelMaximal idealdefinitionField, definition, no evidence levelFielddefinitionPolynomial evaluation, definition, no evidence levelPolynomial evaluationdefinitionIntegral domain, definition, no evidence levelIntegral domaindefinitionExistence of maximal ideals, statement, no evidence levelExistence of maximal …statementRoot of a polynomial, definition, no evidence levelRoot of a polynomialdefinitionSubfield, definition, no evidence levelSubfielddefinitionIrreducible element, definition, no evidence levelIrreducible elementdefinitionField extension, definition, no evidence levelField extensiondefinitionSplit polynomial, definition, no evidence levelSplit polynomialdefinitionIrreducible polynomial, definition, no evidence levelIrreducible polynomialdefinitionDegree of a field extension, definition, no evidence levelDegree of a field ext…definitionAlgebraic element, definition, no evidence levelAlgebraic elementdefinitionIntermediate field, definition, no evidence levelIntermediate fielddefinitionAlgebraically closed field, definition, no evidence levelAlgebraically closed …definitionFinite 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…statementAlgebraic closure, definition, no evidence levelAlgebraic closuredefinitionAn algebraic extension of an algebraic extension is algebraic, statement, no evidence levelAn algebraic extensio…statementExistence of an algebraic closure, statement, no evidence levelExistence of an algeb…statement
Each node uses the nodes above it. The colour of the bar on a box shows the evidence level, and the box also gives it as text. The list below has the same nodes.
  • no evidence level
  1. Group concept
  2. Ring definition
  3. Commutative ring definition
  4. Ideal definition
  5. Polynomial ring definition
  6. Subring definition
  7. Unit of a ring definition
  8. Zero divisor definition
  9. Maximal ideal definition
  10. Field definition
  11. Polynomial evaluation definition
  12. Integral domain definition
  13. Existence of maximal ideals statement
  14. Root of a polynomial definition
  15. Subfield definition
  16. Irreducible element definition
  17. Field extension definition
  18. Split polynomial definition
  19. Irreducible polynomial definition
  20. Degree of a field extension definition
  21. Algebraic element definition
  22. Intermediate field definition
  23. Algebraically closed field definition
  24. Finite extension definition
  25. Algebraic extension definition
  26. Tower law statement
  27. Finite extensions are algebraic statement
  28. Algebraic closure definition
  29. An algebraic extension of an algebraic extension is algebraic statement
  30. Existence of an algebraic closure statement

Sources

  1. Web https://en.wikipedia.org/wiki/Algebraic_closure, Existence of an algebraic closure and splitting fields

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