Classifying Z_2(\alpha) and Z_2(\alpha)^* Groups

[ehrenfest]

[SOLVED] extension field

Homework Statement

Let E be an extension field of Z_2 and $\alpha$ in E be algebraic of degree 3 over Z_2. Classify the groups $<Z_2(\alpha),+>$ and $<Z_2(\alpha)^*,\cdot>$ according to the fundamental theorem of finitely generated abelian groups.
$Z_2(\alpha)^*$ denotes the nonzero elements of Z_2(\alpha).

Homework Equations

The Attempt at a Solution

The first group is obviously Z_2 cross Z_2 cross Z_2, right? I am using that theorem that says that every element of F(\alpha) can be uniquely expressed as a polynomial in F[\alpha] with degree less than 3. I am so confused about how to find the second group since they didn't give me explicitly the irreducible polynomial for $\alpha$ over F? Is the problem impossible?

 

[morphism]

The first group is indeed (Z_2)^3.

As for the second one: How many elements are in (Z_2(alpha))*?

 

[ehrenfest]

8-1=7, so it has to be Z_7!