## Field

### Set

context | $X$ |

definiendum | $\langle X,+,* \rangle \in \mathrm{field}(X)$ |

postulate | $\langle X,+,* \rangle \in \mathrm{divisionRing}(X)$ |

postulate | $\langle X,* \rangle \in \mathrm{abelianGroup}(X)$ |

### 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: Field