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splitting field (Definition)

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

  1. $ f$ splits (factors into a product of linear factors) in $ K[x]$,
  2. $ 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.



"splitting field" is owned by djao.
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See Also: normal extension


Attachments:
splitting field of a finite set of polynomials (Theorem) by polarbear
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Cross-references: ground field, normal extension, isomorphic, property, product, factors, field extension, field, polynomial
There are 28 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 6844 times total.

Classification:
AMS MSC: 12F05 (Field theory and polynomials :: Field extensions :: Algebraic extensions)
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State of eucation by akdevaraj on 2008-05-15 01:52:21

 A few months ago I had been to my alma mater, a college in Mumbai, mainly to to meet one of the faculty members who is well-versed in Latex. He introduced me to a colleague. I was surprised to find that, although she teaches linear algebra or calculus she is totally ignorant of number theory; she had not heard of Fermat's theorem or Euler's totient. Q: Does such a situation prevail in USA?
Devaraj
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