Count & Describe Automorphisms of Z_3 (Set of Integers Modulo 3)

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Discussion Overview

The discussion revolves around the question of counting and describing the automorphisms of the vector space Z_3(alpha), where Z_3 represents the set of integers modulo 3. Participants explore the nature of alpha and its implications for the structure of the vector space, including its dimensionality and the associated invertible matrices.

Discussion Character

  • Homework-related
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • One participant initially expresses uncertainty about the definition of alpha and its relevance to the problem.
  • Another participant suggests that understanding the definition of alpha is crucial for progressing with the question.
  • A later reply clarifies that alpha is defined as a root of the polynomial x^2 + 1, which is significant for determining the structure of Z_3[alpha].
  • It is noted that Z_3[alpha] forms a 2-dimensional vector space over Z_3, leading to the consideration of invertible 2x2 matrices as part of the solution.

Areas of Agreement / Disagreement

Participants generally agree on the importance of defining alpha for the discussion, but there is no consensus on the specific number of automorphisms or their descriptions, as this remains unresolved.

Contextual Notes

The discussion is limited by the initial ambiguity surrounding the definition of alpha and its implications for the vector space structure. The exploration of invertible matrices is contingent on the understanding of the dimensionality of Z_3[alpha].

erraticimpulse
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Please bear with me as I don't have latex. This is a homework question I have and I don't even know if it makes sense:

How many Z_3 (set of integers modulo 3) vector space automorphisms of Z_3(alpha) are there? Describe them.

I'm not sure if alpha is supposed to be the root of some polynomial or just an element outside of Z_3. I know that Z_3 is isomorphic to GF(3) (the Galois field of order 3). Any help would be much appreciated.
 
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If you don't know what alpha is, then we have no chance of knowing what it is. The only person who can help you here is you: find out from your book, notes, the question sheet, what alpha is.
 
Okay well, I figured it out. Alpha was defined a few pages earlier as a root of x^2+1 (a polynomial with coeff's in Z_3[x]). Thanks for pointing out what should have been obvious to me.
 
So Z_3[alpha] is just a 2 dimensional vector space over Z_3, so we're just looking at the invertible 2x2 matrices over Z_3. What answer did you get?
 

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