List the elements of the field F = F3[x] / < x2 + 1 >

In summary, the conversation discusses the deconstruction and elements of a set F3[X], which consists of polynomials with coefficients in F3. One element in the quotient of F3[X] is shown to be f(x) + <x^2+1>, and research shows that there are nine elements in this quotient. The conversation then moves on to discussing polynomial long division and finding the maximum degree of the remainder, r(x). Finally, the conversation mentions showing the equivalence of f(x) and r(x) with the addition of <x^2+1> and how this can provide a hint for solving the problem.
  • #1
MupptMath
4
0
Can someone please explain the deconstruction and elements of this set. I understand it to be..

F3[X] = {f(x)=a(0)+a(1)X+...+a(n)X^n : a(i) in F3]
<x^2+1> = {g(x)(x^2+1): g(x) in F3[x]]

So an element in the quotient should be something like f(x)+<x^2+1>

Yet, research shows there are nine elements:
[0], [1], [2], [x], [x+1], [x+2], [2x], [2x+1], [2x+2]

I just don't see how they are derived.

Thanks!
 
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  • #2
Write $f(x) = q(x)(x^2 + 1) + r(x)$ using polynomial long division.

What can you say about the maximum degree of $r(x)$?

Show $r(x) + \langle x^2 + 1\rangle = f(x) + \langle x^2 + 1\rangle$.

This entails showing $f(x) - r(x)$ is a multiple of $x^2 + 1$.

Does this give you a hint?
 

FAQ: List the elements of the field F = F3[x] / < x2 + 1 >

1. What is the field F3[x]?

F3[x] is a field of polynomials with coefficients in the finite field F3, which contains the elements 0, 1, and 2. This means that all polynomials in F3[x] have coefficients that are either 0, 1, or 2, and all arithmetic operations on these polynomials follow the rules of finite fields.

2. What does the notation < x2 + 1 > mean?

The notation < x2 + 1 > represents the ideal generated by the polynomial x2 + 1. This means that all elements in the ideal are multiples of x2 + 1, and any polynomial in F3[x] can be divided by x2 + 1 to yield a quotient and remainder.

3. How is the field F = F3[x] / < x2 + 1 > constructed?

The field F = F3[x] / < x2 + 1 > is constructed by taking the set of all polynomials in F3[x], and factoring out the ideal < x2 + 1 >. This results in a new set of elements, called cosets, which are the remainders of the polynomials when divided by x2 + 1. These cosets form the elements of the field F.

4. What is the significance of the polynomial x2 + 1 in this field construction?

The polynomial x2 + 1 is called the irreducible polynomial in F3[x], which means it cannot be factored into smaller polynomials with coefficients in F3. This is important in constructing the field F = F3[x] / < x2 + 1 >, as it ensures that the elements of this field are unique and have no further divisors.

5. How does the field F = F3[x] / < x2 + 1 > differ from the field F3?

The field F = F3[x] / < x2 + 1 > is an extension of the field F3, meaning it contains all the elements of F3 and additional elements created by factoring out the ideal < x2 + 1 >. These additional elements are necessary for certain mathematical operations, such as finding roots of polynomials, that cannot be performed in the field F3 alone.

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