Math 2.0

Field theory and polynomials Definition

Prime subfield

Builds on FieldSubfieldCharacteristic of a ring

Definition

Let KK be a field. The prime subfield of KK is the intersection of all the subfields of KK. It is the smallest subfield of KK.

Remarks

If the characteristic of KK is a prime pp, then the prime subfield is isomorphic to 𝔽p=ℤ/pℤ\mathbb{F}_p = \mathbb{Z}/p\mathbb{Z}. If the characteristic of KK is 00, then the prime subfield is isomorphic to ℚ\mathbb{Q}. The prime subfield of ℝ\mathbb{R} is ℚ\mathbb{Q}.

Reading path

To understand this page from the ground up, read these 9 pages in order. The foundations come first.

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Group, concept, no evidence levelGroupconceptRing, definition, no evidence levelRingdefinitionCharacteristic of a ring, definition, no evidence levelCharacteristic of a r…definitionCommutative 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 levelSubfielddefinitionPrime subfield, definition, no evidence levelPrime subfielddefinition
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. Characteristic of a ring definition
  4. Commutative ring definition
  5. Subring definition
  6. Unit of a ring definition
  7. Field definition
  8. Subfield definition
  9. Prime subfield definition

Where this leads

6 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Field_(mathematics), Subfields and prime fields
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DefinesPrime subfield
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