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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
- 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
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 20 results over 5 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click 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)