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

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In summary, a factor ring, also known as a quotient ring, is formed by dividing a ring by one of its ideals. It is represented as R/I, where R is the original ring and I is the ideal. The notation F[x]/<x^3+X+1> & F[x]/<x^3+X^2+1> represents factor rings formed by dividing the ring of polynomials over field F by the respective ideals, and are important in the study of polynomial rings. The elements of a factor ring are determined by taking the elements of the original ring and adding, subtracting, and multiplying them by the elements of the ideal. Factor rings are related to polynomial rings as they provide insight into the structure
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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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  • #2
The easiest way would be to notice that both are 8-element fields.
 
  • #3
I tried but I couldn't find them, what are the eight?
 
  • #4
Which ones did you find?
 

1. What is a factor ring?

A factor ring, also known as a quotient ring, is a mathematical construct that is formed by dividing a ring by one of its ideals. It consists of the elements of the original ring that are not contained in the ideal. In other words, it is a "simpler" version of the original ring.

2. How do you represent a factor ring?

A factor ring is typically represented as R/I, where R is the original ring and I is the ideal being used to form the factor ring. In the case of F[x]/ & F[x]/, F[x] is the original ring and and are the ideals being used to form the factor rings.

3. What is the significance of the notation F[x]/ & F[x]/?

The notation F[x]/ & F[x]/ represents the factor rings formed by dividing the ring of polynomials over field F by the respective ideals. These factor rings are important in abstract algebra as they allow for the study of properties of polynomials and their roots.

4. How do you determine the elements of a factor ring?

The elements of a factor ring are determined by taking the elements of the original ring and adding, subtracting, and multiplying them by the elements of the ideal. In the case of F[x]/ & F[x]/, the elements of the factor rings would consist of polynomials with coefficients in F that are not divisible by x^3+X+1 and x^3+X^2+1, respectively.

5. What is the relationship between factor rings and polynomial rings?

Factor rings are a type of quotient ring that is formed by dividing a polynomial ring by an ideal. They are important in the study of polynomial rings as they help in understanding the structure and properties of polynomials. In the example of F[x]/ & F[x]/, the factor rings provide insight into the roots and factorization of polynomials over field F.

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