Math 2.0

Field theory and polynomials Field extensions Theorem

Tower law

Builds on Degree of a field extensionIntermediate field

Statement

Let K⊆L⊆MK \subseteq L \subseteq M be fields. Then [M:K]=[M:L]⋅[L:K].[M : K] = [M : L] \cdot [L : K].

Remarks

The proof: let u1,…,udu_1, \dots, u_d be a basis of LL over KK, and let w1,…,wew_1, \dots, w_e be a basis of MM over LL. Then the products uiwju_i w_j are a basis of MM over KK. The formula also holds for infinite degrees, as a product of cardinals. So M/KM/K is finite if and only if M/LM/L and L/KL/K are finite. If [M:K][M : K] is a prime number, then M/KM/K has no intermediate field other than KK and MM.

Reading path

To understand this page from the ground up, read these 11 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 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…definitionIntermediate field, definition, no evidence levelIntermediate fielddefinitionTower law, statement, no evidence levelTower lawstatement
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. 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. Intermediate field definition
  11. Tower law statement

Where this leads

16 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Degree_of_a_field_extension, The multiplicativity formula for degrees
  2. Web https://en.wikipedia.org/wiki/Degree_of_a_field_extension, Proof of the multiplicativity formula in the finite case
About this page
Tag0A7KJ68XHT
URIm2://0A7KJ68XHT
Aliasesmultiplicativity-formula-for-degrees
Kindstatement, theorem
WritingW0 stub
MSC12F05
StatusThis copy of the site has no build data. Run m2 status 0A7KJ68XHT.
UsesDegree of a field extension, Intermediate field

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