Properties of ordered fields Ordered field

the property



a
>
0
∧
x
<
y
⇒
a
x
<
a
y


{\displaystyle a>0\land x<y\rightarrow ax<ay}





the property



x
<
y
⇒
a
+
x
<
a
+
y


{\displaystyle x<y\rightarrow a+x<a+y}

for every a, b, c, d in f:

either −a ≤ 0 ≤ or ≤ 0 ≤ −a
one can add inequalities : if ≤ b , c ≤ d, + c ≤ b + d
one can multiply inequalities positive elements : if ≤ b , 0 ≤ c, ac ≤ bc
transitivity of inequality: if < b , b < c, < c
if x < y , x, y > 0, 1/y < 1/x
1 positive
an ordered field has characteristic 0. (since 1 > 0, 1 + 1 > 0, , 1 + 1 + 1 > 0, etc. if field had characteristic p > 0, −1 sum of p − 1 ones, −1 not positive.) in particular, finite fields cannot ordered.
squares non-negative: 0 ≤ in f

every subfield of ordered field ordered field (inheriting induced ordering). smallest subfield isomorphic rationals (as other field of characteristic 0), , order on rational subfield same order of rationals themselves. if every element of ordered field lies between 2 elements of rational subfield, field said archimedean. otherwise, such field non-archimedean ordered field , contains infinitesimals. example, real numbers form archimedean field, hyperreal numbers form non-archimedean field, because extends real numbers elements greater standard natural number.

an ordered field k isomorphic real number field if every non-empty subset of k upper bound in k has least upper bound in k. property implies field archimedean.

vector spaces on ordered field

vector spaces (particularly, n-spaces) on ordered field exhibit special properties , have specific structures, namely: orientation, convexity, , positively-definite inner product. see real coordinate space#geometric properties , uses discussion of properties of r, can generalized vector spaces on other ordered fields.