Math 2.0

Field theory and polynomials Definition

Algebraic element

Builds on Field extensionRoot of a polynomialPolynomial ring

Definition

Let L/KL/K be a field extension and let a∈La \in L. The element aa is algebraic over KK if there is a nonzero polynomial f∈K[x]f \in K[x] with f(a)=0f(a) = 0.

Remarks

The number 2\sqrt{2} is algebraic over ℚ\mathbb{Q}, because it is a root of x2−2x^2 - 2. Every element a∈Ka \in K is algebraic over KK, as a root of x−ax - a. The element aa is algebraic over KK if and only if [K(a):K][K(a) : K] is finite. An element that is not algebraic is transcendental.

Reading path

To understand this page from the ground up, read these 12 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 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.
  • 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

Where this leads

73 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Algebraic_element, lead section
About this page
Tag0NPW1DWECM
URIm2://0NPW1DWECM
Kinddefinition
WritingW1 stated
MSC12F05
DefinesAlgebraic element
UsesField extension, Root of a polynomial, Polynomial ring

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