A prime field, usually written as Fp or GF(p), contains exactly p elements, where p is prime. Addition, subtraction, multiplication, and division by nonzero elements are performed modulo p. The prime condition is what guarantees that every nonzero element has a multiplicative inverse.
Why prime fields matter
Prime fields are the simplest finite fields and the foundation for many larger constructions. They appear in elliptic-curve cryptography, error-correcting codes, polynomial algorithms, zero-knowledge systems, and computer algebra. Understanding them turns modular arithmetic from a collection of tricks into a coherent algebraic structure.
What this course covers
- How modular arithmetic wraps integers into a finite set.
- The field axioms and why composite moduli create zero divisors.
- How the extended Euclidean algorithm finds multiplicative inverses.
- Fermat's little theorem and fast modular exponentiation.
- Multiplicative order, generators, and primitive roots.
- How to solve linear and quadratic equations modulo a prime.
Prime modulus versus composite modulus
Modulo 7, every nonzero residue has an inverse: for example, 3 × 5 ≡ 1 (mod 7). Modulo 8, however, 2 × 4 ≡ 0 even though neither factor is zero. These zero divisors prevent division from working consistently, so the integers modulo 8 form a ring but not a field.
How to use the interactive lesson
Work through the modules in order, but change the supplied values whenever a calculator appears. Testing examples and counterexamples is the fastest way to see which facts depend on primality and which hold for modular arithmetic more generally.