Simple Extensions

Abstract Algebra · Axiom Academy

LESSON Simple Field Extensions Understanding F(α): The smallest field containing F and an element α, and its structure as a quotient ring. Think of F(α) as the result of "adjoining" α to F. It contains: All elements needed to maintain field properties (closure under +, ×, inverses) 2. Algebraic vs Transcendental Elements The structure of F(α) depends critically on whether α satisfies a polynomial equation over F: For the remainder of this lesson, we focus on the algebraic case , which has rich structure connected to polynomial rings. Key properties of the minimal polynomial: Irreducible: m(x) cannot be factored into non-constant polynomials in F[x] Divides all annihilators: If p(α) = 0, then m(x) | p(x) Unique: There is exactly one such monic polynomial Here's the beautiful connection: when α is algebraic over F with minimal polynomial m(x), we have an isomorphism: This says F(α) is isomorphic to the quotient ring of polynomials modulo the ideal generated by m(x). Why this works: The evaluation homomorphism φ: F[x] → F(α) defined by φ(p(x)) = p(α) has kernel ⟨m(x)⟩, and by the first isomorphism theorem, F[x]/⟨m(x)⟩ ≅ Im(φ) = F(α). If m(x) has degree n, then 1, α, α², ..., α n-1 forms a basis for F(α) over F.

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