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Conjugate algebraic elements over a field
Definition
Let and be elements of field extensions of , both algebraic over . They are conjugate over when they have the same minimal polynomial over (The evaluation kernel and the unique monic irreducible minimal polynomial of an algebraic element), or equivalently when is a root of the minimal polynomial of . The relation is relative to the chosen base field. Relative embeddings and automorphisms are those of -homomorphisms and -embeddings of field extensions.
Depends on
Used by
- Assuming Choice, conjugates in an algebraic closure are related by a base automorphism Corollary
- Conjugates of an average of roots of unity Lemma
- A base-field embedding carries an algebraic element to a conjugate Proposition
- A monic irreducible of degree d over F_q has the d distinct roots α,α^q,…,α^qᵈ⁻¹ Theorem
Dependency tree · two levels
7 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
- P. L. Clark, Field Theory, Chapters 3 to 5 (standard reference, not scraped)
- J. S. Milne, Fields and Galois Theory, Chapters 2, 3, and 5 (standard reference, not scraped)