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This questions is a follow up - but not a duplicate - of this post

Why is $X^4+1$ reducible over $\mathbb F_p$ with $p \geq 3,$ prime

Let $F $ be a field of characteristic $p >0$. Show that $f = t^4+1 \in F[t] $ is not irreducible. Let $K$ be a splitting field of $f $ over $F$. Determine which finite field $F $ must contain so that $K=F $.

As mentioned before, the first part of the question is already answered. But how can I answer the 2nd one, especially as $f $ can have sifferent factorizations?

Squirrel-Power
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1 Answers1

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This is a corollary to Don Antonio's answer to your previous question. The answer is $\Bbb{F}_p$ if $p=2$ or $p\equiv 1 \pmod8$ and $\Bbb{F}_{p^2}$ otherwise, since (for $p$ odd) it's always the smallest extension field that contains primitive 8th roots of unity, which for a finite field is equivalent to having size congruent to 1 mod 8 since its non-zero elements form a cyclic group under multiplication.

If $p$ is even, then $p=2$ and you can do that case by inspection.

C Monsour
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