===== Field =====
==== Set ====
| @#55CCEE: context     | @#55CCEE: $X$ |
| @#FFBB00: definiendum | @#FFBB00: $\langle X,+,* \rangle \in \mathrm{field}(X)$ |
| @#55EE55: postulate   | @#55EE55: $\langle X,+,* \rangle \in \mathrm{divisionRing}(X)$ |
| @#55EE55: postulate   | @#55EE55: $\langle X,* \rangle \in \mathrm{abelianGroup}(X)$ |

==== Discussion ====
A field is essentially two //compatible abelian// groups over a set $X$, one of which is necessarily commutative. Compatible in the sense of the distributive laws of a ring, which is asymmetrical with respect to "$+$" and "$*$".

The second requirement destinguishes the division ring from a division ring by requiring commutivity and of the multiplication $*$. 

One generally (also) calls $F$ the field.

=== Theorems ===
Finite fields are completely determined by their cardinality.

There is a field of cardinality for each $p^n$, $p$ prime.

==== Reference ====
Wikipedia: [[http://en.wikipedia.org/wiki/Field_%28mathematics%29|Field]]
==== Parents ====
=== Subset of ===
[[Division ring]]