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Algebraic Extensions
Definition: Types of Extensions Let: 1. E be an extension field of a field F 2. a ∊ E If: ′a′ is the zero of some nonzero polynomial in F[x] then ′a′ is algebraic over F Else: ′a′ is transcendental over F
If: every element of E is algebraic over F, then: an extension E of F is called an algebraic extension over F Else: E is transcendental extension of F
An extension of F of the form F(a) is called a simple extension of F
Note: F(x) = Field of Quotients of F[x] = {f(x) / g(x) f(x), g(x) ∊ F[x], g(x) ≠0}
Theorem 21.1 Characterization of Extensions Let: 1. E be an extension field of the field F 2. a ∊ E