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

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SUMMARY

The discussion centers on the equivalence of the number of elements in the quotient rings A/ and A/ where A is the integers modulo 7, and b(x) = x^3 - 2 and c(x) = x^3 + 2 are polynomials in A[x]. It is established that both A/ and A/ are fields, indicating that and are maximal ideals. The key conclusion is that since both quotient rings are fields, they have the same number of elements, specifically 7^3, corresponding to the degree of the polynomials.

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  • Understanding of field theory and maximal ideals
  • Familiarity with polynomial rings, specifically A[x]
  • Knowledge of quotient rings and their properties
  • Basic concepts of modular arithmetic, particularly integers modulo 7
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  • Learn about the structure of finite fields and their applications
  • Explore the concept of quotient rings in abstract algebra
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Mathematics students, particularly those studying abstract algebra, algebraic structures, and anyone interested in the properties of polynomial rings and fields.

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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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you stated it wrong it seems. you want A[x]/<whatever>. then it gets easier.
 

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