# Finite Fields of Order 3

I have been trying to learn Real Analysis, though I'm having trouble with a problem.

1. Show that there exists one and (essentially) only one field with three elements.

Any help will be appreciated.

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Maybe you can construct an isomorphism between any two such. – Andrew Nov 9 '12 at 22:42
How did this question appear in real analysis? As for your question every field is an additive group. How many groups do you know with 3 elements? (Upto isomorphism) – fretty Nov 9 '12 at 22:52

When you see a claim in mathematics that says "There exists a unique $X$ up to isomorphism", or "There is one, and (essentially) only one $X$", this means that you are required to prove two things:
1. There is some $X$ with the wanted properties and structure.
2. If $Y$ also has the wanted properties then there is an isomorphism between $X$ and $Y$.
In the case of a field with three elements, you really have little choice of how to define everything. Show that you can do that on the three elements $\{0,1,2\}$, where $0$ is the additive neutral and $1$ is the multiplicative neutral (those have to exist anyway). Now given a field of three elements $\{a,b,c\}$ you have to send the additive neutral to zero, the multiplicative neutral to $1$ and the third element to...?