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 If: ′a′ is transcendental over F ⇒ F(a) ≈ F(x) If: ′a′ is algebraic over F ⇒ F(a) ≈ F[x] / <p(x)> p(x) = poly. in F[x] of minimum degree such that: 1. p(a) = 0 and 2. p(x) is irreducible over F Proof; Consider: homomorphism φ: F[x] → F(a), given by f(x) → f(a) If: ′a′ is transcendental over F ⇒ Kerφ = {0} ⇒ Can extend φ to an isomorphism ψ: F(x) → F(a) such that ψ(f(x)/g(x)) = f(a)/g(a) If: ′a′ is algebraic over F: ⇒ Kerφ ≠ {0} Theorem 16.4: there is a polynomial p(x) ∊ F[x] such that: 1. Kerφ = <p(x)> and 2. p(x) has a minimum degree among all the nonzero elements of Kerφ ⇒ p(a) = 0 and since p(x) is a polynomial of min. degree with this property, it is irreducible over F Theorem 21.2 Uniqueness Property If a is algebraic over a field F, then there is a unique monic irreducible polynomial p(x) in F[x] such that p(a) = 0 Polynomial is with this property is called the minimal polynomial for ‘a’ over F Theorem 21.3 Divisibility Property Let: 1. ‘a’ be algebraic over F 2. p(x) be the minimal polynomial for ‘a’ over F If: f(x) ∊ F[x] and f(a) = 0 Then: p(x) divides f(x) in F[x] Definition: Degree of an Extension Let E be an extension field of a field F E has degree n over F if E has dimension ‘n’ as a vector space over F (Notation: [E:F] = n) If: [E:F] is finite, E is called a finite extension of F Else: E is an infinite extension of F Theorem 21.4 Finite Implies Algebraic If E is a finite extension of F, then E is an algebraic extension of F Theorem 21.5 [K:F] = [K:E][E:F] Let: 1. K be a finite extension field of the field E 2. E be a finite extension field of the field F Then: K is a finite extension field of F and [K:E][E:F] Theorem 21.6 Primitive Element Theorem If F is a field of characteristic 0, and ‘a’ and ‘b’ are algebraic over F Then: there is an element c in F(a,b) such that F(a,b) = F(c) Theorem 21.7 Algebraic over Algebraic is Algebraic If: 1. K is an algebraic extension of E and 2. E is an algebraic extension of F Then: K is an algebraic extension of F Corollary: Subfield of Algebraic Elements Let E be an extension field of the field F. Then the set of all elements of E that are algebraic over F is a subfield of E .