A field **(F. +, *)** is a set F with two operations + and * (usually called addition and multiplication), such that the following axioms hold:

**(F, +)**is an**abelian group**(identity element 0)**(F\{0}, *)**is an**abelian group**(identity element 1)**Distributivity**of multiplication over addition: For all*a*,*b*and*c*in*F*, the following equality holds:

*a*· (*b*+*c*) = (*a*·*b*) + (*a*·*c*)

Another definition:

A field is a nonzero commutative ring that contains a multiplicative inverse for every nonzero element.

- An infinite field extension has an infinite number of intermediate fields
- Roots and multiplicity of the polynomial x
^{3}– 2 - Roots and multiplicity of the polynomial x
^{5}– x - Every finite integral domain is a field
- Q(i) and Q(√2) are isomorphic as verctor spaces, but not as fields
- Field extension and multiplicativity formula for degrees