(a) Show that the polynomial $ f(x)=x^{3}+x+1 $ is irreducible over $ \mathbb{Z}_{5}[x] $ . (b) If a root $ \xi $ of the polynomial f(x) is adjoined to $ \mathbb{Z}_{5} $ , how many elements are there in the resulting field $ \mathbb{Z}_{5}(\xi) $. $$ $$ I got 125 elements. Is it true ?
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Yes, it is true: $\mathbf Z_5(\xi)$ is a $\mathbf Z_5$-vector space of dimension $3$. – Bernard Apr 25 '17 at 22:25
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Because $f$ is irreducible $\mathbb{Z}_5(\xi) \simeq \mathbb{Z}_5[x]/(f(x))$ – reuns Apr 25 '17 at 22:39