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How can I factorize $X^p+1$ in a field of characteristic $p$ ?

If the field is $\mathbb Z/p\mathbb Z$, then $$X^p+1=X^p+1^p=(X+1)^p,$$ but if the field is $\mathbb F_{p^n}$ or if the characteristic is $q\neq p$, I don't know how to do it.

It's not duplicate, but it solve my problem for $\mathbb F_{p^n}$. But what happen for a field of characteristic $q\neq p$ ?

Jannes Braet
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user386627
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    It's still true for a general field of characteristic $p$ that $(X + 1)^p = X^p + 1$. – Travis Willse May 04 '17 at 10:59
  • Nothing happens at all in general if $p \neq q$. The left hand side might even be irreducible in this case, so you can't find such a nice factorization in general. – Dirk May 04 '17 at 11:03
  • Factorizing $X^p+1$ in a field of characteristic $q \neq p$ depends on both $q$ and the actual field..I mean you can always take $\overline{\mathbb F_q}$ and then it factors into distinct linear factors... – MooS May 04 '17 at 11:03
  • If you want to consider char $p$, it is a duplicate, if you want to consider char $\neq p$ it is unclear what you are asking. Pick your poison, this gets closed anyway. – MooS May 04 '17 at 11:06

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