Math 2.0

statement theorem

An algebraic extension of an algebraic extension is algebraic

Statement

Let K⊆L⊆MK \subseteq L \subseteq M be fields. If M/LM/L and L/KL/K are algebraic, then M/KM/K is algebraic.

Remarks

The proof: let a∈Ma \in M have minimal polynomial xn+cn−1xn−1+⋯+c0x^n + c_{n-1} x^{n-1} + \dots + c_0 over LL. The field E=K(c0,…,cn−1)E = K(c_0, \dots, c_{n-1}) is finite over KK, and E(a)E(a) is finite over EE. By the tower law, E(a)/KE(a)/K is finite, so aa is algebraic over KK. Also, if M/KM/K is algebraic, then M/LM/L and L/KL/K are algebraic.

Explanations

L1 human

Algebraic extensions can be stacked. If each step adds only algebraic elements, then the whole tower adds only algebraic elements.

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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…statementAn algebraic extension of an algebraic extension is algebraic, statement, no evidence levelAn algebraic extensio…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. Intermediate field definition
  15. Finite extension definition
  16. Algebraic extension definition
  17. Tower law statement
  18. Finite extensions are algebraic statement
  19. An algebraic extension of an algebraic extension is algebraic statement

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Sources

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

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