- #1

Bashyboy

- 1,421

- 5

## Homework Statement

Show that any two Fields with four elements are isomorphic.

## Homework Equations

## The Attempt at a Solution

Let ##F= \{0,1,a,b\}## be a field with four elements. Since ##F## is an additive abelian group, by Lagrange's theorem ##1## must have either an order of ##2## or ##4##; i.e., either ##1+1=0## or ##1+1+1+1=0##. Since ##F## is a field, every nonzero element is a unit, meaning that the group of units ##U(F)## is a group of ##3## which makes it isomorphic to ##\Bbb{Z}_3##. Note that neither ##a+b=a## nor ##a+b=b## can hold. This leaves us with ##a+b = 0## or ##a+b=1##...This is where I get stuck. I am trying to show that ##1+1 = 0##. If I can rule out ##a+b=1##, then I can show ##b=-a##, and therefore ##(-a)^3 = 1## or ##-1=1## or ##1+1=0##. Another route is to assume that ##1+1+1+1=0## is true, which would imply ##F \simeq \Bbb{Z}_4##, and then deduce a contradiction, but I haven't been able to identify the contradiction. Perhaps there is a problem with ##F \simeq \Bbb{Z}_4## and ##U(F) \simeq \Bbb{Z}_3## being simultaneously true...I could use some guidance.