Find a number that is algebraic with degree 3 over Z_3

[ehrenfest]

Homework Statement

I want to find a number that is algebraic with degree 3 over Z_3. To do this, I need to find an extension field of Z_3. Q,R, and Z_p (p greater than 3) definitely will not work because they have different algebra. Anyone?

Homework Equations

The Attempt at a Solution

 

[Hurkyl]

Homework Statement

I want to find a number that is algebraic with degree 3 over Z_3. To do this, I need to find an extension field of Z_3. Q,R, and Z_p (p greater than 3) definitely will not work because they have different algebra. Anyone?

How about the algebraic closure of Z_3? Or is that too nonconstructuve?

Well, at least we know that no matter what extension field E you use and number $\alpha$ you select, there has to be a map $\pi : (\mathbb{Z} / 3\mathbb{Z})[t] \to E$ with $\pi(t) = \alpha$.