Math 2.0

Field theory and polynomials Definition

Finite extension

Builds on Degree of a field extension

Definition

A field extension L/KL/K is finite if its degree [L:K][L : K] is finite. Otherwise L/KL/K is an infinite extension.

Remarks

A finite extension is not the same as an extension of finite fields. For example, ℂ/ℝ\mathbb{C}/\mathbb{R} is finite, and both fields are infinite. The extension ℚ(23)/ℚ\mathbb{Q}(\sqrt[3]{2})/\mathbb{Q} is finite of degree 33. The field K(x)K(x) of rational functions is an infinite extension of KK, because 1,x,x2,…1, x, x^2, \dots are linearly independent over KK. Finite extensions are algebraic.

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Group, concept, no evidence levelGroupconceptRing, definition, no evidence levelRingdefinitionCommutative ring, definition, no evidence levelCommutative ringdefinitionSubring, definition, no evidence levelSubringdefinitionUnit of a ring, definition, no evidence levelUnit of a ringdefinitionField, definition, no evidence levelFielddefinitionSubfield, definition, no evidence levelSubfielddefinitionField extension, definition, no evidence levelField extensiondefinitionDegree of a field extension, definition, no evidence levelDegree of a field ext…definitionFinite extension, definition, no evidence levelFinite extensiondefinition
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  1. Group concept
  2. Ring definition
  3. Commutative ring definition
  4. Subring definition
  5. Unit of a ring definition
  6. Field definition
  7. Subfield definition
  8. Field extension definition
  9. Degree of a field extension definition
  10. Finite extension definition

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16 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Degree_of_a_field_extension, Definition and notation
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