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field (Definition)

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. $ a+(-a) = 0$ for some element $ -a \in F$ (existence of additive inverses)
  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$.



"field" is owned by djao.
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Attachments:
examples of fields (Example) by AxelBoldt
characterization of field (Theorem) by alozano
division (Definition) by pahio
groups in field (Topic) by pahio
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Cross-references: identity, commutative ring, distributive property, multiplicative inverses, unity, associativity of multiplication, inverses, additive, zero element, commutativity, associativity, properties, multiplication, addition, binary operations
There are 721 references to this entry.

This is version 4 of field, born on 2001-10-19, modified 2008-05-08.
Object id is 355, canonical name is Field.
Accessed 33583 times total.

Classification:
AMS MSC: 12E99 (Field theory and polynomials :: General field theory :: Miscellaneous)
Pending Errata and Addenda
None.
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Discussion
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linguistic whine by rspuzio on 2007-03-16 01:08:26
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 Indo-European 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?
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