[SOLVED] extension field 1. The problem statement, all variables and given/known data Let E be an extension field of Z_2 and [itex]\alpha[/itex] in E be algebraic of degree 3 over Z_2. Classify the groups [itex]<Z_2(\alpha),+>[/itex] and [itex]<Z_2(\alpha)^*,\cdot>[/itex] according to the fundamental theorem of finitely generated abelian groups. [itex]Z_2(\alpha)^*[/itex] denotes the nonzero elements of Z_2(\alpha). 2. Relevant equations 3. 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 [itex]\alpha[/itex] over F? Is the problem impossible?