Math 2.0

Field theory and polynomials Definition

Normal extension

Builds on Algebraic extensionIrreducible polynomialSplit polynomial

Definition

Let L/KL/K be an algebraic extension. The extension L/KL/K is normal if every irreducible polynomial f∈K[x]f \in K[x] that has a root in LL splits over LL.

Remarks

A finite extension L/KL/K is normal if and only if LL is the splitting field of some polynomial over KK. Every extension of degree 22 is normal. For example, ℚ(2)/ℚ\mathbb{Q}(\sqrt{2})/\mathbb{Q} is normal. The extension ℚ(23)/ℚ\mathbb{Q}(\sqrt[3]{2})/\mathbb{Q} is not normal: x3−2x^3 - 2 has a root in it, but its other two roots are not real. Normality is not transitive: ℚ(24)/ℚ(2)\mathbb{Q}(\sqrt[4]{2})/\mathbb{Q}(\sqrt{2}) and ℚ(2)/ℚ\mathbb{Q}(\sqrt{2})/\mathbb{Q} are normal, but ℚ(24)/ℚ\mathbb{Q}(\sqrt[4]{2})/\mathbb{Q} is not.

Reading path

To understand this page from the ground up, read these 19 pages in order. The foundations come first.

Show the path as a graph
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 ringdefinitionZero divisor, definition, no evidence levelZero divisordefinitionPolynomial evaluation, definition, no evidence levelPolynomial evaluationdefinitionField, definition, no evidence levelFielddefinitionIntegral domain, definition, no evidence levelIntegral domaindefinitionRoot of a polynomial, definition, no evidence levelRoot of a polynomialdefinitionSubfield, definition, no evidence levelSubfielddefinitionIrreducible element, definition, no evidence levelIrreducible elementdefinitionSplit polynomial, definition, no evidence levelSplit polynomialdefinitionField extension, definition, no evidence levelField extensiondefinitionIrreducible polynomial, definition, no evidence levelIrreducible polynomialdefinitionAlgebraic element, definition, no evidence levelAlgebraic elementdefinitionAlgebraic extension, definition, no evidence levelAlgebraic extensiondefinitionNormal extension, definition, no evidence levelNormal extensiondefinition
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. Zero divisor definition
  8. Polynomial evaluation definition
  9. Field definition
  10. Integral domain definition
  11. Root of a polynomial definition
  12. Subfield definition
  13. Irreducible element definition
  14. Split polynomial definition
  15. Field extension definition
  16. Irreducible polynomial definition
  17. Algebraic element definition
  18. Algebraic extension definition
  19. Normal extension definition

Where this leads

17 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Normal_extension, Definition
About this page
Tag0CBRFQ7DVZ
URIm2://0CBRFQ7DVZ
Kinddefinition
WritingW1 stated
MSC12F10
DefinesNormal extension
UsesAlgebraic extension, Irreducible polynomial, Split polynomial

Markdown · JSON · Edit on GitHub

What this page needs for W2

  • edge instance_of in: one example (0 of 1)
  • edge not_instance_of in: one non-example that fails one clause (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

Used by