Tuesday, 13 December 2016

abstract algebra - How many elements are in the quotient ring fracmathbbZ3[x]langle2x3+x+1rangle




How many elements are in the quotient ring Z3[x]2x3+x+1 ?



I guess I should be using the division algorithm but I'm stuck on how to figure it out.


Answer



Any element of this quotient ring is of the form a0+a1x+a2x2+2x3+x+1 where aiZ3



Thus we have 27 elements



Note:whenever there is a polynomial f(x) of degree 3Z3[x] then by division algorithm we will get two polynomials q(x),r(x) such that f(x)=(2x3+x+1)q(x)+r(x) where r(x)=0 or degr(x)<deg(2x3+x+1) which is 2



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