Formal Root of a function to describe a Finite Field

FoundBread

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Aug 1, 2018
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Hi I'm doing coding theory and am unsure what exactly I am being asked.
Was asked to show that f(x) = x4 + x3 + 2x + 1 is irreducible in Z3[x] which is fine.

But then asked to use a formal root of f(x), α, to describe the finite field F81.

I think I understand how to construct a field of order 81 by finding an irreducible polynomial of degree 2 over Z9[X] since 92 = 81...

But I'm not sure what exactly is meant by using a formal root of f(x) to describe the field.

Any help would be much appreciated,
Thanks
 
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