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ehrenfest
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[SOLVED] extension field
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).
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?
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?
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