Math 2.0

Field theory and polynomials Definition

Field generated by adjoining elements

Builds on Field extensionSubfieldIntermediate field

Definition

Let L/KL/K be a field extension and let S⊆LS \subseteq L. The field K(S)K(S) is the intersection of all the subfields of LL that contain KK and SS. It is the smallest intermediate field of L/KL/K that contains SS.

Remarks

For S={a1,…,an}S = \{a_1, \dots, a_n\} one writes K(a1,…,an)K(a_1, \dots, a_n). The elements of K(S)K(S) are the quotients f(s1,…,sn)/g(s1,…,sn)f(s_1, \dots, s_n)/g(s_1, \dots, s_n) with f,gf, g polynomials over KK, si∈Ss_i \in S and g(s1,…,sn)≠0g(s_1, \dots, s_n) \neq 0. One has K(a,b)=K(a)(b)K(a, b) = K(a)(b). For example, ℚ(2)={p+q2:p,q∈ℚ}\mathbb{Q}(\sqrt{2}) = \{ p + q \sqrt{2} : p, q \in \mathbb{Q} \} inside ℝ\mathbb{R}.

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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 extensiondefinitionIntermediate field, definition, no evidence levelIntermediate fielddefinitionField generated by adjoining elements, definition, no evidence levelField generated by ad…definition
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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. Intermediate field definition
  10. Field generated by adjoining elements definition

Where this leads

37 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Field_extension, Extension field
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WritingW0 stub
MSC12F05
DefinesField generated by adjoining elements
UsesField extension, Subfield, Intermediate field

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