Fields
★★★★☆Undergraduate
📖Definition
A field is an algebraic structure where all four arithmetic operations are freely possible. It's a commutative ring where every non-zero element has a multiplicative inverse.
📐Formulas
(F, +, ·) is a commutative ring with unity
Commutative ring with unity
∀ a ≠ 0, ∃ a⁻¹ : a · a⁻¹ = 1
Multiplicative inverse for non-zero elements
char(F) = 0 or prime p
Characteristic of field
✏️Examples
예제 1
Describe operations in finite field 𝔽₅ = {0, 1, 2, 3, 4}.
예제 2
Show ℚ(√2) is a field.
📜History
Discovered by: Leopold Kronecker, Richard Dedekind (19th century)
Field concept emerged from studying algebraic equations and developing algebraic number theory.
⚡Applications
Cryptography
Finite field crypto (AES, ECC)
Coding Theory
Error-correcting codes
Algebraic Geometry
Algebraic varieties
🔗Related Documents
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