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

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In summary, the conversation discusses classifying the groups <Z_2(\alpha),+> and <Z_2(\alpha)^*,\cdot> based on the fundamental theorem of finitely generated abelian groups, where Z_2(\alpha) is an extension field of Z_2 and \alpha is algebraic of degree 3 over Z_2. The first group is identified as (Z_2)^3 and the second group is determined to have 7 elements, making it isomorphic to Z_7. The challenge of finding the second group without the explicitly given irreducible polynomial for \alpha is acknowledged.
  • #1
ehrenfest
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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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  • #2
The first group is indeed (Z_2)^3.

As for the second one: How many elements are in (Z_2(alpha))*?
 
  • #3
8-1=7, so it has to be Z_7!
 

1. What is a Z2(α) group?

A Z2(α) group is a mathematical group consisting of elements that can be divided into two distinct subsets, with the operation of the group being defined as multiplication. The group is denoted by Z2(α) and is commonly used in abstract algebra and group theory.

2. What is the significance of the α in Z2(α)?

The α in Z2(α) represents a specific element within the group and is often referred to as a generator. It is used to distinguish between different Z2(α) groups and plays a crucial role in classifying them.

3. How are Z2(α) and Z2(α)* groups related?

Z2(α) and Z2(α)* groups are closely related, with Z2(α)* being the dual group of Z2(α). This means that the elements of Z2(α)* are functions that map the elements of Z2(α) to complex numbers, and the operation of the group is defined as pointwise multiplication of these functions.

4. How are Z2(α) and Z2(α)* groups classified?

Z2(α) and Z2(α)* groups can be classified based on their order, which is the number of elements in the group. They can also be classified based on their structure, such as whether they are cyclic, abelian, or non-abelian.

5. What are some applications of Z2(α) and Z2(α)* groups?

Z2(α) and Z2(α)* groups have various applications in mathematics, physics, and computer science. They are used in cryptography, coding theory, and quantum computing. They also have applications in understanding the structure of molecules and in studying symmetries in physical systems.

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