Math 2.0

statement proposition

Finite extensions are algebraic

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.

Explanations

L1 human

In a finite-dimensional extension, the powers of an element cannot all be independent. A dependence between the powers is a polynomial equation for the element.

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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
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  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

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Sources

  1. Web https://en.wikipedia.org/wiki/Algebraic_extension, Some properties

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