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I have this problem and don't know how to solve it. Any one can help? Thanks

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closed as off-topic by Daniel, Leucippus, user91500, HK Lee, JonMark Perry Mar 25 '17 at 9:52

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Take $\mathbb{F}_5[x] / (p(x))$ where $p(x)$ is an irreducible polynomial of degree $3$.

One example of an irreducible polynomial of degree $3$ in $\mathbb{F}_5[x]$ is $x^3 + x + 1$. We can check this by evaluating that $x^3 + x + 1$ has no roots.

$\mathbb{F}_5 [x]$ is a PID since $\mathbb{F}_5$ is a field. Since $p(x)$ is irreducible in $\mathbb{F}_5[x]$, $\mathbb{F}_5[x] / (p(x))$ is a field.

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  • $\begingroup$ A quadratic polynomial with degree $3$? $\endgroup$ – Bernard Mar 24 '17 at 23:54
  • $\begingroup$ Sorry bad mistake. $\endgroup$ – Dean Young Mar 24 '17 at 23:56
  • $\begingroup$ This is the canonical answer for constructing finite fields of a given order. $\endgroup$ – ChocolateAndCheese Mar 25 '17 at 4:22

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