Do A/<b(x)> and A/<c(x)> have the same number of elements?

In summary, a polynomial factor ring is a quotient ring created by dividing a polynomial ring by a specific polynomial. Its purpose is to simplify and solve equations involving polynomials, and it differs from a polynomial ring in that it is a smaller structure contained within the larger polynomial ring. The process of creating a polynomial factor ring involves choosing a polynomial to divide the polynomial ring by and using the division algorithm. Some practical applications of polynomial factor rings include solving systems of equations, coding theory, cryptography, and signal processing.
  • #1
bigreddog
1
0
Let A be the integers modulo 7.
b(x)= x^3 -2 and c(x) = x^3 + 2 are polynomials in A[x].

How can you show that A/<b(x)> and A/<c(x)> have the same number of elements? In this practice problem I already showed that A/<b(x)> and A/<c(x)> are fields by showing that <b(x)> and <c(x)> are maximal ideals, but I don't know how this helps.
 
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  • #2
you stated it wrong it seems. you want A[x]/<whatever>. then it gets easier.
 

FAQ: Do A/<b(x)> and A/<c(x)> have the same number of elements?

What is a polynomial factor ring?

A polynomial factor ring is a mathematical structure that is created by taking a polynomial ring and dividing it by a specific polynomial. It is also known as a quotient ring.

What is the purpose of polynomial factor rings?

The purpose of polynomial factor rings is to simplify and solve equations involving polynomials. By dividing a polynomial ring by a specific polynomial, we can reduce the complexity of the equation and make it easier to solve.

How are polynomial factor rings different from polynomial rings?

Polynomial factor rings are created by dividing a polynomial ring, while a polynomial ring is a set of polynomials with operations such as addition and multiplication defined on them. A polynomial ring is a larger structure that contains polynomial factor rings.

What is the process of creating a polynomial factor ring?

To create a polynomial factor ring, we start with a polynomial ring and choose a polynomial to divide it by. We then use the division algorithm to divide the polynomial ring by the chosen polynomial, resulting in a new structure with elements that are the remainders of the division.

What are some practical applications of polynomial factor rings?

Polynomial factor rings have various applications in mathematics, computer science, and engineering. They are used in solving systems of equations, coding theory, cryptography, and signal processing, among others.

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