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A cyclic extension is a finite Galois extension with cyclic Galois group
Definition
Let be a finite Galois extension (Finite Galois extensions and ). It is cyclic when its Galois group is a cyclic group.
When is generated by a specific automorphism , that generator may be named in the body of a later theorem or example, but it is not part of the data of the definition.
Depends on
Used by
- The Lagrange resolvent attached to a cyclic action and a root of unity Definition
- Additive Hilbert 90: trace zero is the image of α↦α-σ(α) Theorem
- Hilbert's theorem 90 for a finite cyclic extension Theorem
- In characteristic p, a degree-p extension is cyclic exactly when it is generated by a root of xᵖ-x-a with a∈ F and that polynomial irreducible Theorem
Dependency tree · two levels
4 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- J. S. Milne, Fields and Galois Theory, v5.10, Chapter 6, Section 6 (standard reference, not scraped)
- J. Ash, Basic Abstract Algebra, Section 6.7 (standard reference, not scraped)