Proving Z₅ is a Field

Abstract Algebra · Axiom Academy

Verify field properties by constructing the multiplication table and finding all multiplicative inverses Excellent work! You've successfully proven that Z₅ is a field. Here's what we established: Field Requirements: A field must satisfy many axioms, but the crucial one for finite sets is that every non-zero element must have a multiplicative inverse. Multiplication Table Method: By constructing the complete multiplication table, we can systematically find inverses by looking for 1 in each row. All Inverses Found: We verified that 1⁻¹ = 1, 2⁻¹ = 3, 3⁻¹ = 2, and 4⁻¹ = 4. Every non-zero element has an inverse! Why This Works: Z₅ is a field because 5 is prime. For any prime p , Z_p forms a field. If n is composite, Z_n is only a ring, not a field. Practical Importance: Finite fields like Z₅ are fundamental in cryptography, coding theory, and error correction. This verification technique applies to any Z_p where p is prime. Try constructing the multiplication table for Z₇ or Z₁₁ to see the pattern!

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