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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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    This is analogous to constructing the complex numbers over the reals, based on the irreducible polynomial $x^2+1$. Think how this is used to define the complex numbers. Then do something similar in your case.2012-11-25
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    Your field will be $\mathbb{Z}_7[x]/\langle x^3 + x + 1 \rangle$. That means you can identify elements of the field with polynomials of degree $\le 2$, and the laws are given by adding or multiplying elements and then reducing them modulo $x^3 + x + 1$ (that is computing the remainder in the division).2012-11-25

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