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- Jan 26, 2012

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**Problem**: Let $D$ be an integral domain. A polynomial $f(x)$ from $D[x]$ that is neither the zero polynomial nor a unit in $D[x]$ is said to be i

*rreducible over $D$*if, whenever $f(x)$ is expressed as a product $f(x)=g(x)h(x)$, with $g(x),h(x)\in D[x]$, then $g(x)$ or $h(x)$ is a unit in $D[x]$.

(a) Suppose that $f(x)\in\mathbb{Z}_p[x]$ and is irreducible over $\mathbb{Z}_p$, where $p$ is prime. If $\deg f(x)=n$, prove that $\mathbb{Z}_p[X]/\langle f(x)\rangle$ is a field with $p^n$ elements.

(b) Construct a field with 27 elements.

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