Constructing Q(√2)

Abstract Algebra · Axiom Academy

Adjoining √2 to the rationals to create a 2-dimensional vector space over Q with field operations Excellent work! You've explored the construction of Q(√2) as a field extension. Here's what we learned: Field Extension Structure: Q(√2) = a + b√2 | a, b ∈ Q is a 2-dimensional vector space over Q Basis and Uniqueness: The basis 1, √2 spans Q(√2), and every element has a unique representation as a + b√2 Closure Under Operations: Addition and multiplication preserve the form a + b√2 with rational coefficients Rationalization Technique: Division requires multiplying by the conjugate (a - b√2) to eliminate irrationals from denominators Field Properties: Q(√2) forms a field—it's closed under addition, multiplication, and has multiplicative inverses (via rationalization) This construction generalizes! For any algebraic number α, we can form Q(α) similarly. The dimension [Q(α):Q] equals the degree of α's minimal polynomial. In our case, √2 satisfies x² - 2 = 0, giving us dimension 2.

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