Field Extensions
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| Extension Field | A field E is an EXTENSION FIELD of a field F, if F (subset of) E and the operations of F are the same as the operations of E.
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| Examples of Extension Fields | R is an extension of Q
C is an extension of R
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| Fundamental Theorem of Field Theory | Let F be a field and f(x) is a non-constant polynomial in F[x], there there exists an extension field E in which f(x) has a root
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| Fundamental Theorem of Field Theory Example | f(x)=x^2-2 in Q. f has no roots in Q but has a root in R
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| ~field of quotients theorem | If R is an integral domain, then it is contained in it's field of quotients.
EX. {a/b|a,b in Z, b~=0}
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| Def: Split | Let f(x) be in F[x]. We say f(x) splits in an extension field E of F if we can write f(x)=(x-a1)(x-a2)...(x-an) where ai is in E not necessarily distinct. i.e. f can be factored into a product of linear factors in E[x].
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| Def: Splitting field | E is called a SPLITTING FIELD for f(x) over f, if f(x) splits in E[x] but has no proper subfield of E.
The splitting field for f(x) is the smallest extension of F in which f(x) splits.
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| Theorem about splitting fields | Splitting fields always exist and are unique up to isomorphism
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| Irreducible Theorem | Let F be a field and p(x) in F[x] is irreducible over F. If a is a root of p(x) in some extension E of F, then F(a) is isomorphic to F[x]/<p(x)>.
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