Field Axioms and Examples

Abstract Algebra · Axiom Academy

LESSON Field Axioms and Examples Understanding the complete algebraic structure of fields, with examples from rational numbers to finite fields. (F1) Closure: For all a, b ∈ F, both a + b and a × b are in F (F2) Associativity: (a + b) + c = a + (b + c) and (a × b) × c = a × (b × c) (F3) Commutativity: a + b = b + a and a × b = b × a (F4) Additive Identity: ∃ 0 ∈ F such that a + 0 = a for all a (F5) Multiplicative Identity: ∃ 1 ∈ F (1 ≠ 0) such that a × 1 = a for all a (F6) Additive Inverses: For all a ∈ F, ∃ (-a) ∈ F such that a + (-a) = 0 (F7) Multiplicative Inverses: For all a ∈ F with a ≠ 0, ∃ a⁻¹ ∈ F such that a × a⁻¹ = 1 (F8) Distributivity: a × (b + c) = (a × b) + (a × c) 2. Standard Examples of Fields Let's examine the most important examples of fields in mathematics: Rational Functions: The set F(x) of all rational functions p(x)/q(x) where p, q are polynomials over a field F and q ≠ 0 also forms a field. Let's verify that ℤ₅ = 0, 1, 2, 3, 4 with addition and multiplication mod 5 satisfies all field axioms. The crucial property is that every nonzero element has a multiplicative inverse. 4 × 4 = 16 = 1 mod 5 → 4⁻¹ = 4 Consider ℤ₆ = 0, 1, 2, 3, 4, 5 with arithmetic mod 6. Although it has addition and multiplication, it fails to be a field. The deeper issue: 6 = 2 × 3 is composite, so ℤ₆ has zero divisors . Notice that 2 × 3 = 0 mod 6, even though neither 2 nor 3 is zero. 5. Fields vs. Integral Domains

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