Math 2.0

Field theory and polynomials Proposition

Finite extensions are algebraic

Builds on Finite extensionAlgebraic extension

Statement

Let L/KL/K be a finite extension. Then L/KL/K is algebraic.

Remarks

The proof: let n=[L:K]n = [L : K] and a∈La \in L. The n+1n + 1 elements 1,a,…,an1, a, \dots, a^n are linearly dependent over KK. A linear dependence c0+c1a+⋯+cnan=0c_0 + c_1 a + \dots + c_n a^n = 0 gives a nonzero polynomial with root aa. So every element of LL has degree at most [L:K][L : K] over KK. The converse is false, as the algebraic numbers show. An extension generated by finitely many algebraic elements is finite, by the tower law.

Reading path

To understand this page from the ground up, read these 16 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 elementdefinitionFinite extension, definition, no evidence levelFinite extensiondefinitionAlgebraic extension, definition, no evidence levelAlgebraic extensiondefinitionFinite extensions are algebraic, statement, no evidence levelFinite extensions are…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. Finite extension definition
  15. Algebraic extension definition
  16. Finite extensions are algebraic statement

Where this leads

3 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Algebraic_extension, Some properties
About this page
Tag0DFYTNZ6AM
URIm2://0DFYTNZ6AM
Kindstatement, proposition
WritingW0 stub
MSC12F05
StatusThis copy of the site has no build data. Run m2 status 0DFYTNZ6AM.
UsesFinite extension, Algebraic extension

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