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May 10th, 2017, 09:56 PM   #1
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Irreducibility and splitting fields of a polynomial

If i'm given a polynomial $x^3 + 2x + 5$ how can I tell if it's irreducible over $\mathbb{Q}[x]$? I've tried Eisenstiens irreducibility criterion with no luck. Also if we can't factor a polynomial, is its splitting field just
$ \mathbb{F} \backslash <x^3 + 2x + 5 > $ ?
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June 4th, 2017, 07:56 AM   #2
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If the polynomial were reducible over $\displaystyle \mathbb{Q},$ it would be reducible over $\displaystyle \mathbb{Z}.$
And it would be reducible modulo $\displaystyle 3$. But it doesn't have a root in $\displaystyle \mathbb{F}_3.$ That means it is irreducible over $\displaystyle \mathbb{Q}.$

Using a computer one can see that it has one real root and two complex non-real roots. It means field $\displaystyle \mathbb{Q}[x]/<x^3 + 2x + 5>$ is not the splitting field.
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