A field is a commutative ring with identity F = (F, + ,−,0,·,1
) such that
F is non-trivial: 0 ≠ 1 and
every non-zero element has a multiplicative inverse: x ≠ 0 ⇒ ∃y (x·y = 1).
Remark: The inverse of x is unique, and is usually denoted by x−1.
Let F and G be fields. A morphism from F to G is a function h : F→G that is a homomorphism: h(x + y) = h(x) + h(y) and h(x·y) = h(x)·h(y) and h(1) = 1.
Remark: It follows that h(0) = 0 and h(−x) = −h(x).
0 is a zero for ·: 0·x = x and x·0 = 0.
(Q, + ,−,0,·,1), the field of rational numbers with addition, subtraction, zero, multiplication, and one.
|Congruence n-permutable||yes, n = 2|
|Congruence extension property|
|Definable principal congruences|
|Equationally definable principal congruences|
|Strong amalgamation property|
|Epimorphisms are surjective|