Algebra Hub
Abstract Algebra // Structures

FIELD
THEORY

In elementary math, we study specific numbers. In Abstract Algebra, we study the rules themselves. A "Field" is the gold standard of algebra—a structure where addition, subtraction, multiplication, and division work perfectly without breaking the system.

The Axioms of a Field

A Field (denoted as F\mathbb{F}) requires a set of elements and two operations: ++ and ×\times. It must satisfy rigorous axioms, the most difficult being the existence of a multiplicative inverse for every non-zero element.

∀a≠0∈F,∃a−1 such that a⋅a−1=1\forall a \neq 0 \in \mathbb{F}, \exists a^{-1} \text{ such that } a \cdot a^{-1} = 1

The Rational numbers (Q\mathbb{Q}) are a field. The Integers (Z\mathbb{Z}) are not a field, because the multiplicative inverse of 22 is 1/21/2, which is not an integer.

Finite Fields (Galois Fields)

Fields do not have to be infinite. We can create finite fields using modular ("clock") arithmetic, denoted as GF(p)GF(p). But there is a massive catch: the modulus must be a prime number.

If you use a composite number (like modulo 6), the system breaks. You get Zero Divisors. For example, 2×3≡0(mod6)2 \times 3 \equiv 0 \pmod{6}. If you can multiply two non-zero numbers and get zero, you can no longer divide safely!

Use the generator to the right. Set the modulus to 5 (Prime) and notice how every row has a glowing 1 (an inverse). Then set it to 6 (Composite) and watch the red 0 anomalies destroy the field!

Cayley Table Engine

Finite Field Modulo N

Valid Field GF(p)

Because 5 is prime, every non-zero element has a multiplicative inverse (a '1' in its row). Division is safe.

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Hover over the matrix to view calculations.

Abstract Structures

Ascend the hierarchy of abstract algebraic structures.

Group Theory

The mathematical study of symmetry. Rubik’s cubes, crystallography, and particle physics.

Ring Theory

Structures with addition and multiplication, but where division might not be guaranteed.

Galois Theory

The beautiful bridge connecting Field Theory to Group Theory. Proving why quintic equations cannot be solved.