FielDs

Fields

A field is a [[ring]] in which * is commutative and the nonzero elements form a [[group]] under *.

A more useable definition is the following:

The field (R,+,*) contains a set (R) and two binary operators (+,*) such that:

# + is associative:  a+(b+c) = (a+b)+c
# + has an "zero" z, written 0, such that a+z = z+a = a
# for each a in R there is an opposite -a such that a + (-a) = (-a) + a = 0
# for all a, b in R: a+b = b+a
# * is associative: a*(b*c) = (a*b)*c
# * has an identity e, written 1, such that for all non-zero a: a * 1 = 1 * a = a
# each non-zero element a has an inverse, inv(a), such that a * inv(a) = inv(a) * a = 1
# for all a and b, a*b = b*a
# for all a,b,c: a*(b+c) = a*b + a*c and (b+c)*a = b*a + c*a