Understanding Factor Rings F[x]/<x^3+X+1> & F[x]/<x^3+X^2+1>

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Homework Help Overview

The discussion revolves around understanding the structure of the factor rings F[x]/ and F[x]/, where F is Z mod 2. The original poster expresses confusion about how to approach the problem and what elements these factor rings contain.

Discussion Character

  • Exploratory, Conceptual clarification

Approaches and Questions Raised

  • The original poster seeks to understand the nature of the factor rings and their elements. Some participants suggest that recognizing both as 8-element fields could be a starting point, while others inquire about specific elements that have been identified.

Discussion Status

The discussion is ongoing, with participants exploring the properties of the factor rings and attempting to clarify their understanding. There is an indication that some guidance has been provided regarding the number of elements in the rings, but no consensus has been reached on the specific elements themselves.

Contextual Notes

The original poster is working under the constraints of a homework assignment, which may limit the resources they can use to explore the problem further.

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I'm lost and don't even know where to start.

Let F = Z mod 2, show F[x]/<x^3+X+1> and F[x]/<x^3+X^2+1> are isomorphic.

I guess fist I need help understanding what those two factor rings look like and what elements they contain.
Thanks
 
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The easiest way would be to notice that both are 8-element fields.
 
I tried but I couldn't find them, what are the eight?
 
Which ones did you find?
 

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