field
A field is a number system where all four arithmetic operations — add, subtract, multiply, and divide (except by zero) — always work and stay inside the set. The rational numbers and the real numbers are fields; this is the structure that makes high-school algebra run smoothly, because you can solve a · x = b simply by dividing whenever a is not zero.
Formally, a field is a commutative ring with identity in which every nonzero element has a multiplicative inverse. Equivalently, the nonzero elements form an abelian group under multiplication. That single added guarantee — invertibility of every nonzero element — is what separates a field from a mere integral domain like the integers.
Division by zero remains forbidden even in a field: 0 has no multiplicative inverse, since 0 times anything is 0, never 1. Fields come in many flavours: the rationals, the reals, the complex numbers, and the finite fields with a prime number of elements (the integers mod p). They are the natural home for solving equations, building vector spaces, and the whole of Galois theory.
The integers mod 5, {0, 1, 2, 3, 4}, form a field because 5 is prime: every nonzero element has an inverse, e.g. 2 · 3 = 6 ≡ 1, so 2⁻¹ = 3. By contrast the integers mod 6 are not a field, since 2 has no inverse there.
ℤ/5ℤ is a field (5 prime); ℤ/6ℤ is not.
“Field” in the algebraic sense (a number system) is unrelated to the “field” of a database record or a vector field in physics — same English word, different meaning. Every field is an integral domain, and every finite integral domain is in fact a field.