(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 ?
2026-03-12 21:15:37.1773350137
If a root $ \xi $ of the polynomial f(x) is adjoined to $ \mathbb{Z}_{5} $
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