# Explain how to obtain a field with $125$ elements using polynomials by using concrete example provided by a polynomial [closed]

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

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

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Daniel, Leucippus, user91500, HK Lee, JonMark Perry
If this question can be reworded to fit the rules in the help center, please edit the question.

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.
• A quadratic polynomial with degree $3$? – Bernard Mar 24 '17 at 23:54