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A normal algebraic extension is one in which every minimal polynomial with a root in the extension splits there
Definition
An algebraic field extension is normal if, for every , the minimal polynomial of over splits over .
Equivalently, every irreducible polynomial that has one root in splits over . Indeed, the monic associate of such a is the minimal polynomial of any one of its roots in , and multiplying by a nonzero scalar does not change whether a polynomial splits.
Depends on
Used by
- Every purely inseparable algebraic extension is normal Corollary
- ℚ(³√2)/ℚ is separable and nonnormal with trivial automorphism group Counterexample
- Finite Galois extensions and Gal(K/F) Definition
- The normal closure of an algebraic extension inside a fixed algebraic closure Definition
- A finite normal extension is separable over its purely inseparable fixed field Lemma
- Conjugates of an average of roots of unity Lemma
- A nonempty intersection of normal subextensions inside a common algebraic extension is normal Proposition
- A normal extension generated by finitely many elements is the splitting field of the product of their minimal polynomials Proposition
- An algebraic extension that is a splitting field of a polynomial is normal Proposition
- If K/F is normal and F⊆ E⊆ K, then K/E is normal Proposition
- An algebraic extension generated by elements whose minimal polynomials split in it is normal Theorem
- Normal subgroups, conjugate fields, and quotient groups in the Galois correspondence Theorem
- Polynomial algebras over fields have finite integral closures Theorem
Dependency tree · two levels
8 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
- The Stacks Project, Section 9.15: Normal extensions (standard reference, not scraped)
- J. S. Milne, Fields and Galois Theory, Chapter 2 (standard reference, not scraped)