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Find the degree of the splitting field of $f(x):=x^3-5$ over $F:=\mathbb{F}_7$.

Attempt:

$f$ is irreduicible in $F[x]$ (suppose in contradiction it is reducible, thus it splits to at least one linear normalized polynomial element but this is contradiction beacuse $f$ has no roots in $F$). Thus, if $E$ is the splitting field of $f$ over $F$, we get $F\subset F(5^{1\over3})\subseteq E$. So I get that the degree is at least $2$.

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If $[E:F]=2$ and $\alpha^3=5$ then there exists some polynomial $g$ of degree $2$ such that $g(\alpha)=0$. But then $$f(x)=g(x)q(x)+r(x)$$ for some polynomials $q$ and $r$, and the degree of $r$ is at most $1$. Therefore $$0=f(\alpha)=q(\alpha)q(\alpha)+r(\alpha)=r(\alpha)$$ So $r=0$. This is a contradiction because $f$ is irreducible.

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  • $\begingroup$ Why does $|E:F|=2$ imply the existence of $g$? $\endgroup$ – J. Doe May 2 at 15:17
  • $\begingroup$ It is the very definition of the degree of an extension. $\endgroup$ – ajotatxe May 2 at 17:48
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Hint: $f(x)=x^3-5$ is irreducible over $\Bbb F_7$ since it has no root.

Further reference:

Splitting field of $x^3 - 2$ over $\mathbb{F}_5$

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  • $\begingroup$ I noticed that but not sure how it helps. $\endgroup$ – J. Doe May 2 at 14:54
  • $\begingroup$ And if $\alpha$ is one root then $\alpha^7$ is another... $\endgroup$ – Jyrki Lahtonen May 2 at 14:55
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Hint:

If $\omega$ is a root of $x^3-5$ in some extension (it has no root in $\mathbf F_7$), search for cube roots in $\mathbf F_7$. Deduce the splitting field is $\mathbf F_7[\omega]$.

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