Journal of Formalized Mathematics
Volume 2, 1990
University of Bialystok
Copyright (c) 1990 Association of Mizar Users

## Properties of Fields

Jozef Bialas
University of Lodz
Supported by RPBP.III-24.C9.

### Summary.

The second part of considerations concerning groups and fields. It includes a definition and properties of commutative field $F$ as a structure defined by: the set, a support of $F$, containing two different elements, by two binary operations ${\bf +}_{F}$, ${\bf \cdot}_{F}$ on this set, called addition and multiplication, and by two elements from the support of $F$, ${\bf 0}_{F}$ being neutral for addition and ${\bf 1}_{F}$ being neutral for multiplication. This structure is named a field if $\langle$the support of $F$, ${\bf +}_{F}$, ${\bf 0}_{F} \rangle$ and $\langle$the support of $F$, ${\bf \cdot}_{F}$, ${\bf 1}_{F} \rangle$ are commutative groups and multiplication has the property of left-hand and right-hand distributivity with respect to addition. It is demonstrated that the field $F$ satisfies the definition of a field in the axiomatic approach.

#### MML Identifier: REALSET2

The terminology and notation used in this paper have been introduced in the following articles [6] [4] [8] [9] [2] [3] [7] [5] [1]

Contents (PDF format)

#### Bibliography

[1] Jozef Bialas. Group and field definitions. Journal of Formalized Mathematics, 1, 1989.
[2] Czeslaw Bylinski. Functions and their basic properties. Journal of Formalized Mathematics, 1, 1989.
[3] Czeslaw Bylinski. Functions from a set to a set. Journal of Formalized Mathematics, 1, 1989.
[4] Czeslaw Bylinski. Some basic properties of sets. Journal of Formalized Mathematics, 1, 1989.
[5] Eugeniusz Kusak, Wojciech Leonczuk, and Michal Muzalewski. Abelian groups, fields and vector spaces. Journal of Formalized Mathematics, 1, 1989.
[6] Andrzej Trybulec. Tarski Grothendieck set theory. Journal of Formalized Mathematics, Axiomatics, 1989.
[7] Wojciech A. Trybulec. Vectors in real linear space. Journal of Formalized Mathematics, 1, 1989.
[8] Zinaida Trybulec. Properties of subsets. Journal of Formalized Mathematics, 1, 1989.
[9] Edmund Woronowicz. Relations and their basic properties. Journal of Formalized Mathematics, 1, 1989.