Math 2.0

Field theory and polynomials Definition

Fixed field

Builds on SubfieldRing isomorphism

Definition

Let LL be a field and let HH be a group of ring automorphisms of LL. The fixed field of HH is LH={a∈L:σ(a)=a for every σ∈H}.L^H = \{ a \in L : \sigma(a) = a \text{ for every } \sigma \in H \}.

Remarks

The set LHL^H is a subfield of LL, because each σ∈H\sigma \in H respects the field operations. For L=ℂL = \mathbb{C} and H={id,σ}H = \{\mathrm{id}, \sigma\}, with σ\sigma complex conjugation, LH=ℝL^H = \mathbb{R}. If HH is finite, then [L:LH]=∣H∣[L : L^H] = |H| and L/LHL/L^H is a Galois extension, by a theorem of Emil Artin. More generally, a group acting on a ring has a fixed subring, the ring of invariants.

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Group, concept, no evidence levelGroupconceptRing, definition, no evidence levelRingdefinitionCommutative ring, definition, no evidence levelCommutative ringdefinitionRing homomorphism, definition, no evidence levelRing homomorphismdefinitionSubring, definition, no evidence levelSubringdefinitionUnit of a ring, definition, no evidence levelUnit of a ringdefinitionRing isomorphism, definition, no evidence levelRing isomorphismdefinitionField, definition, no evidence levelFielddefinitionSubfield, definition, no evidence levelSubfielddefinitionFixed field, definition, no evidence levelFixed fielddefinition
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. Ring homomorphism definition
  5. Subring definition
  6. Unit of a ring definition
  7. Ring isomorphism definition
  8. Field definition
  9. Subfield definition
  10. Fixed field definition

Where this leads

7 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Fixed-point_subring, lead section
  2. Web https://en.wikipedia.org/wiki/Galois_extension, lead section
About this page
Tag0KMJ6919WS
URIm2://0KMJ6919WS
Kinddefinition
WritingW0 stub
MSC12F10
DefinesFixed field
UsesSubfield, Ring isomorphism

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