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

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SUMMARY

The discussion centers on determining the number of automorphisms of the vector space Z_3(α), where α is defined as a root of the polynomial x² + 1 in Z_3[x]. It is established that Z_3 is isomorphic to GF(3), the Galois field of order 3, and Z_3[α] forms a 2-dimensional vector space over Z_3. The focus is on identifying the invertible 2x2 matrices over Z_3, which represent the automorphisms of this vector space.

PREREQUISITES
  • Understanding of Galois fields, specifically GF(3)
  • Knowledge of vector spaces and their dimensions
  • Familiarity with polynomial roots in finite fields
  • Basic linear algebra concepts, particularly invertible matrices
NEXT STEPS
  • Study the properties of Galois fields, focusing on GF(3)
  • Learn about vector space automorphisms and their representations
  • Explore the classification of invertible matrices over finite fields
  • Investigate the implications of polynomial roots in vector spaces
USEFUL FOR

This discussion is beneficial for students studying abstract algebra, particularly those focusing on finite fields and vector spaces, as well as educators and researchers interested in automorphisms and linear transformations in algebraic structures.

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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