Finding a Minimal Polynomial

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Homework Statement


Find the minimal polynomial of a=y^3 in F=Kron(Z/2Z, x^4+x+1). (Calculate the powers of a^2, a^3, and a^4.)

Homework Equations





The Attempt at a Solution



I attempted this trying to follow a similar worked problem in my book:

a=y^3 & y^4=y+1

Multiply by y^-3: y=y^-2 + y^-3

Plug in a

y=a+1

y^4+y+1 = 0 ... Multiply by y^-4: 1+y^-3+y^-4 = 0

Plug in a: a+1+a^-1 = a+1+a^2

So, a satisfies the irreducible polynomial x^2+x+1. Thus, each of the 16 elements of F can be written as a polynomial of degree at most 2 in a and a^2+a+1=0.
So, F=Kron(Z/2Z, a, x^2+x+1)

...did I do this correctly, or am I even close? I'm not sure of the relevance of calculating the powers of a^2, a^3, and a^4 as hinted in the problem statement.
 

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