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splitting field
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(Definition)
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Let $f \in F[x]$ be a polynomial over a field $F$ . A splitting field for $f$ is a field extension $K$ of $F$ such that
- $f$ splits (factors into a product of linear factors) in $K[x]$ ,
- $K$ is the smallest field with this property (any sub-extension field of $K$ which satisfies the first property is equal to $K$ ).
Theorem: Any polynomial over any field has a splitting field, and any two such splitting fields are isomorphic. A splitting field is always a normal extension of the ground field.
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"splitting field" is owned by djao.
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Cross-references: ground field, normal extension, isomorphic, theorem, property, product, factors, field extension, field, polynomial
There are 35 references to this entry.
This is version 3 of splitting field, born on 2002-01-05, modified 2002-11-25.
Object id is 1303, canonical name is SplittingField.
Accessed 8763 times total.
Classification:
| AMS MSC: | 12F05 (Field theory and polynomials :: Field extensions :: Algebraic extensions) |
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Pending Errata and Addenda
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