Math 2.0

Field theory and polynomials Definition

Algebraic extension

Builds on Field extensionAlgebraic element

Definition

A field extension L/KL/K is algebraic if every element of LL is algebraic over KK.

Remarks

Every finite extension is algebraic, so ℂ/ℝ\mathbb{C}/\mathbb{R} and ℚ(2)/ℚ\mathbb{Q}(\sqrt{2})/\mathbb{Q} are algebraic. The converse fails: the field of all algebraic numbers is an algebraic extension of ℚ\mathbb{Q} of infinite degree. The extension ℝ/ℚ\mathbb{R}/\mathbb{Q} is not algebraic, because π\pi is transcendental. An algebraically closed field has no algebraic extension other than itself.

Reading path

To understand this page from the ground up, read these 13 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 extensiondefinitionAlgebraic element, definition, no evidence levelAlgebraic elementdefinitionAlgebraic extension, definition, no evidence levelAlgebraic extensiondefinition
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. Algebraic element definition
  13. Algebraic extension definition

Where this leads

47 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Algebraic_extension, lead section
About this page
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Kinddefinition
WritingW1 stated
MSC12F05
DefinesAlgebraic extension
UsesField extension, Algebraic element

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