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

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SUMMARY

The discussion centers on classifying the groups and for an extension field E of Z_2, where \alpha is algebraic of degree 3 over Z_2. The first group is definitively classified as (Z_2)^3, based on the fundamental theorem of finitely generated abelian groups. The second group, consisting of the nonzero elements , contains 7 elements, leading to the conclusion that it is isomorphic to Z_7.

PREREQUISITES
  • Understanding of extension fields, specifically Z_2 and its algebraic elements.
  • Familiarity with the fundamental theorem of finitely generated abelian groups.
  • Knowledge of polynomial expressions in finite fields.
  • Basic group theory, particularly regarding group classification.
NEXT STEPS
  • Study the structure of finite fields, focusing on Z_2 and its extensions.
  • Learn about the classification of abelian groups using the fundamental theorem.
  • Investigate irreducible polynomials over finite fields and their significance.
  • Explore group isomorphisms, particularly in the context of multiplicative groups of finite fields.
USEFUL FOR

Mathematicians, students in abstract algebra, and anyone interested in the classification of groups in finite fields will benefit from this discussion.

ehrenfest
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[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?
 
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The first group is indeed (Z_2)^3.

As for the second one: How many elements are in (Z_2(alpha))*?
 
8-1=7, so it has to be Z_7!
 

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