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A field is a set $F$ together with two binary operations on $F$, called addition and multiplication, and denoted $+$ and $\cdot$, satisfying the following properties, for all $a,b,c\in F$:
1. $a+(b+c)=(a+b)+c$ (associativity of addition)
2. $a+b=b+a$ (commutativity of addition)
3. $a+0=a$ for some element $0\in F$ (existence of zero element)
4. 5. $a\cdot(b\cdot c)=(a\cdot b)\cdot c$ (associativity of multiplication)
6. $a\cdot b=b\cdot a$ (commutativity of multiplication)
7. $a\cdot 1=a$ for some element $1\in F$, with $1\neq 0$ (existence of unity element)
8. If $a\neq 0$, then $a\cdot a^{{1}}=1$ for some element $a^{{1}}\in F$ (existence of multiplicative inverses)
9. $a\cdot(b+c)=(a\cdot b)+(a\cdot c)$ (distributive property)
Equivalently, a field is a commutative ring $F$ with identity such that:

$1\neq 0$

If $a\in F$, and $a\neq 0$, then there exists $b\in F$ with $a\cdot b=1$.
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Comments
linguistic whine
Why does English have to be cursed with this awful choice of
term "field" for this concept which happend to coincide with
a completely unrelated concept in differential geometry??!!!?
In other languages, the situation is much better  for
instance, in Polish, one says "cialo" for the algebraic
concept and "pole" for the geometric concept; in Greek, one
says "soma" for the algebraic concept and "pedion" for the
geometric concept; in French, one says "corps" for the
algebraic concept and "champs" for the geometric concept....
It would be a lot saner if English followed the pattern
of other IndoEuropean languages and called this thing "body"
and reserved the term "field" for the geometric notion.
By the way, Pahio, how is the situation in the Finnish?
Re: linguistic whine
The situation in the Finnish and also in the other FinnoUgric languages, I think, is quite good and clear.
In Finnish: kunta = 'corps' and kentt\"a = 'champs'
In Hungarian respectively: test and mez\H{o} (o with two acute accents).
The word "kunta" belongs to the oldest FU words (has been at least 6000 years unchanged, but e.g. in Hungarian it has developed to the form "had"!). The present everyday senses of "kunta" are 'community', 'group of people with some structure', 'rural district'.
The everyday senses of "kentt\"a" are 'even area', 'room, space'.
You Rspuzio seem to be a bit worried about the situation of "field" in mathematics. What do you think to do? =o)
Jussi
Re: linguistic whine
I actually thought that algebraic fields actually had a connection to geometry that some people had deliberately sought to obscure, some way to visualize the algebraic concept geometrically that would actually make it easier to understand. But the very mixed parentage of English is just as plausible an explanation for this confusion.
Re: linguistic whine
In french, I think "champs" is now also used for an algebraic stack, so they're not in the clear either.
Cam
Re: linguistic whine
> But the very mixed parentage of English is
> just as plausible an explanation for this confusion.
The Russian is not so mixed language as the English, but the situation is similar: "pole" (i.e. \cyrp\cyro\cyrl\cyre) is used both for 'champs' and 'corps'.