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Let [itex]p[/itex] be a prime and let [itex]x=\zeta_p[/itex] to be a primitive pth root of unity.

How could we conclude

[tex]\frac{\mathbf{Z}[x]/(1-x)}{(x^{p-1}+\cdots+x+1)}=\mathbf{Z}/p\mathbf{Z}=\mathbb{F}_p ?[/tex]

This should be obvious but it seems that I missed something. Could anyone help?

Am I correct to infer that if we want to force [itex]1-x = 0[/itex], i.e. [itex]x = 1[/itex] and hence [itex]\mathbf{Z}[x]/(1-x)=\mathbf{Z}[/itex]. So [itex]\frac{\mathbf{Z}[x]/(1-x)}{(x^{p-1}+\cdots+x+1)}=Z/(p)[/itex] and therefore gives the result?

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# Z[x]/(1-x)/(x^{p-1}+ +x+1) = Z/pZ ?

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