# 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
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## Expert Answer

meldElafellrbo
Answered 2022-02-13 Author has 13 answers
Let $f={x}^{3}+a{x}^{2}+bx+c$ be an irreducible polynomial over ${\mathbb{Z}}_{11}$.
The extension field $E={\mathbb{Z}}_{11}\frac{x}{⟨f⟩}$ contains a zero of f, namely the residue class $\alpha =\stackrel{―}{x}+⟨f⟩$, This gives , and the extension field is $E=\left\{u{\alpha }^{2}+v\alpha +w\mid u,v,w\in {\mathbb{Z}}_{11}\right\}$ with degree $\left[E:{\mathbb{Z}}_{11}\right]=3$. In particular, if f is primitive, the powers of $\alpha$ are exactly the nonzero elements of E.
NB: $f\left(\stackrel{―}{x}\right)=f\left(x+⟨f⟩\right)=f\left(x\right)+⟨f⟩=⟨f⟩=\stackrel{―}{0}$.
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