Explain how to construct a field of order $343$ not using addition and multiplication tables.
I understand that every finite field has order $p^n$ for some prime $p$. Since $343$ is $7^3$, let $p=7$. I believe I need to find a polynomial of degree 3 which does not factor over $\mathbb{Z}_7$. I have considered the following polynomial $x^3+x+1$ and showed that it is irreducible over $\mathbb{Z}_7$.
I am not sure how to procede from here, any help would be great.
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