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Extension field

  • Thread starter ehrenfest
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  • #1
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[SOLVED] extension field

Homework Statement


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).


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 [itex]\alpha[/itex] over F? Is the problem impossible?
 
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Answers and Replies

  • #2
morphism
Science Advisor
Homework Helper
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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))*?
 
  • #3
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8-1=7, so it has to be Z_7!
 

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