Math 2.0

Field theory and polynomials Definition

Transcendental element

Builds on Algebraic element

Definition

Let L/KL/K be a field extension and let a∈La \in L. The element aa is transcendental over KK if aa is not algebraic over KK. That is, f(a)≠0f(a) \neq 0 for every nonzero f∈K[x]f \in K[x].

Remarks

The number π\pi is transcendental over ℚ\mathbb{Q}, but it is algebraic over ℝ\mathbb{R}, as a root of x−πx - \pi. The variable xx is transcendental over KK in the field K(x)K(x) of rational functions. If aa is transcendental over KK, then K(a)K(a) is isomorphic to K(x)K(x), and [K(a):K][K(a) : K] is infinite.

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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 elementdefinitionTranscendental element, definition, no evidence levelTranscendental elementdefinition
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.
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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. Algebraic element definition
  13. Transcendental element definition

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5 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Algebraic_element, Examples
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