General confusion about quotient rings and fields

[samp]

Homework Statement

I'm having a hard time understanding quotient rings. I think an example would help me best understand them.

For example, how does the ring structure of $$\mathbb{F}_{2}/(x^4 + x^2 + 1)$$ differ from that of $$\mathbb{F}_{2}/(x^4 + x + 1)$$?

Homework Equations

The Attempt at a Solution

Are there 16 elements in each of these quotient rings? Since $$2^4 = 16$$. If so, why does the degree of the polynomial determine this? Are either of these fields? I think I need to figure out if the polynomials are irreducible, how do I do this? Do I find all irreducible elements of each quotient ring and try to divide the ideal-generating polynomials by them and see if I get remainder 0? If so, how do I know which elements are irreducible?

Sorry for my stupidity; any help at all would be really appreciated.
Thanks guys,
Sam

 

[mighty2000]

Hello Sam,

Quotient rings can be a difficult concept to grasp, so don't worry about feeling confused. Let's start with a basic definition: a quotient ring is a ring formed by taking a larger ring and dividing it by a smaller ring, known as an ideal. In your first example, we have the ring \mathbb{F}_{2} (the field with 2 elements) and we are dividing it by the ideal (x^4 + x^2 + 1). This means that we are essentially creating a new ring where every element is equivalent to the remainder when divided by (x^4 + x^2 + 1).

To answer your first question, yes, there are 16 elements in each of these quotient rings. This is because the degree of the polynomial (x^4 in this case) determines the number of elements in a finite field. In general, if we have a polynomial of degree n, then the quotient ring will have n^m elements, where m is the number of elements in the base field. In this case, since we are working with the field \mathbb{F}_{2}, which has 2 elements, we have 2^4 = 16 elements.

Now, to answer your question about the difference between the two quotient rings, the main difference lies in the ideal-generating polynomials. In the first quotient ring, (x^4 + x^2 + 1) is not an irreducible polynomial. This means that it can be factored into smaller polynomials. On the other hand, (x^4 + x + 1) is an irreducible polynomial, meaning it cannot be factored into smaller polynomials. This difference will affect the structure of the two quotient rings and the operations that can be performed on their elements.

To determine if a polynomial is irreducible, you can use a few different methods. One method is to try to divide the polynomial by all possible smaller polynomials and see if any of them result in a remainder of 0. If none of them do, then the polynomial is irreducible. Another method is to use the Eisenstein criterion, which states that if a polynomial is irreducible, then it must have a prime number as its leading coefficient and all other coefficients must be divisible by that prime.

I hope this helps clarify the concept of quotient rings for you. Don't worry about feeling stupid, understanding abstract algebra concepts takes time and practice