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Irreducible Polynomial over a Field

Please see the attached file for the fully formatted problems.

5. Find an irreducible polynomial f(x) over the field Z3 with
Z3[x]/(f(x)) = F243.

Note that 243 = 3^5 .

Please explain your reasoning and solution in as much detail as possible.
Thank You.

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Solution Summary

An irreducible polynomial over a given field is found. The solution is detailed and well presented.

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