I think if I show that $x^4+1$ is irreducible in $\Bbb Z_7[x] $, then is maximal in $\Bbb Z_{7}[x]$, but please assure me.
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1Since $Z / 7Z$ is a field, you can indeed try show that the polynomial $x^4+1$ is irreducible to get that $(x^4+1)$ is maximal. – Riquelme May 28 '19 at 06:31
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2try to factor $x^4+1$ into a product of quadratic polynomials – Lozenges May 28 '19 at 06:34
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4See here for a generalization. That polynomial is not irreducible modulo any prime. May be I should vote to close this as a duplicate? – Jyrki Lahtonen May 28 '19 at 06:58