Question

{Abstract Algebra - Ring Theory}

The ideal, I, contains all multiples of x5+x2 1 in the polynomial ring Z2x]. Assuming that x31- 1 (mod I), explain why the order of x in the quotient ring cannot be smaller than 31.

Answer 1

"Z_2 = {0, 1} Z-2 in a field then Z_2 [x] is an integer domain let f(x) = x^5 + x^2 + I then f(0) = I and f9i) = 1 so f(x) is on irreducible polynomial in Z_2 [x] therefore the ideal I = < x^5 + x^2 + i > is a maximal ideal of Z_2 [x] then Z_2 [x]/I is a field with 2^5 = 32 element [by theorem - 1 (below)] now Z_2 [x]/2 = f_2^5 (say) we know that each non zero element in a field is a unit and the set of all unit of a field forms a group under multiplication let U(2^5) be the set of all unit of F_2^5 then U)2^5) is a group under multiplication and