Order of field extension I'm working on some algebra exercises and I'm really struggling with fi

scrimaeua 2022-02-12 Answered
Order of field extension
Im
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meldElafellrbo
Answered 2022-02-13 Author has 13 answers
Let f=x3+ax2+bx+c be an irreducible polynomial over Z11.
The extension field E=Z11xf contains a zero of f, namely the residue class α=x+f, This gives α3+aα2+bα+c=,so α3=aα2bαc, and the extension field is E={uα2+vα+wu,v,wZ11} with degree [E:Z11]=3. In particular, if f is primitive, the powers of α are exactly the nonzero elements of E.
NB: f(x)=f(x+f)=f(x)+f=f=0.
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