The Field of Algebraic Numbers

Abstract Algebra · Axiom Academy

EXAMPLE The Field of Algebraic Numbers Constructing Q̄ and verifying algebraic numbers through minimal polynomials Excellent work! You've explored the field of algebraic numbers Q̄. Here's what we learned: Algebraic Numbers: A complex number is algebraic over ℚ if it's a root of a non-zero polynomial with rational coefficients. Minimal Polynomials: Every algebraic number has a unique monic irreducible polynomial over ℚ of smallest degree - its minimal polynomial. Examples Verified: We showed √2, i, ∛2, √3 are all algebraic by finding their minimal polynomials with degrees 2, 2, 3, and 2 respectively. Field Structure: Q̄ is closed under addition, subtraction, multiplication, and division (by non-zero elements), making it a field. Algebraic Closure: Q̄ is the algebraic closure of ℚ in ℂ - it contains all roots of polynomials with rational coefficients. Not All Complex Numbers: Importantly, Q̄ ⊊ ℂ because transcendental numbers like π and e are not algebraic! This construction is fundamental to algebraic number theory and shows how we can extend ℚ to include solutions to polynomial equations while maintaining field structure. The degree of the minimal polynomial gives us important information about the algebraic number's complexity!

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