Most definitions of "field" I've seen include the requirement 1≠0. One book didn't, and instead defined the "characteristic" of a field as the smallest non-negative integer n such that(adsbygoogle = window.adsbygoogle || []).push({});

[tex]\sum_{k=1}^n 1=0[/tex]

The books that include the requirement 1≠0 then go on to define an "ordered field", and a "complete ordered field". ("Complete" in the sense that every set that has an upper bound has a least upper bound). Then they claim that all complete ordered fields are isomorphic. This seems to overlook the possibility that 1+1=0, or that 1+1+....+1=0 for some number of 1s on the left. Does the definition of a complete ordered field (including the axiom 1≠0) imply that the field has characteristic 0?

**Physics Forums - The Fusion of Science and Community**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# 1+1 = 0

**Physics Forums - The Fusion of Science and Community**