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Here is a test I have taken in my algebra class:

If $f \in \mathbb Z[X]$ and a prime $p$ does not divide the leading coefficient and $\bar{f} \in \mathbb F_p[X]$ is irreducible, then $f$ is irreducible.

And then my professor gave this polynomial $$X^4 - 10 X^2 + 1 $$ saying it is irreducible but its reduction mod p are all reducible and asked us to be caution for this case.

My question is what the name of the first test is (if there is a name for it)?

My question is what is the relation between the test given at the beginning and this polynomial?

Could anyone clarify this to me please?

Edit:

Also, I would like to see some examples of applying this test?

What about if we apply the test when $p = 5$ for this polynomial $X^4 - 10 X^2 + 1 $ ?

Emptymind
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    There is no standard name. The relation is that that the example shows the test is a sufficient but not a necessary criterion for testing irreducibility over $\Bbb Z.\ \ $ – Bill Dubuque Jan 29 '24 at 18:05
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    What Bill said. When $p=5$ the quartic reduces to $x^4+1$. But modulo five $1\equiv-4$, and we have the factorization $$x^4-10x^2+1\equiv x^4-4=(x^2-2)(x^2+2)\pmod5.$$ – Jyrki Lahtonen Jan 30 '24 at 05:32
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    See for example here for a discussion about this particular polynomial. The same polynomial appears in many textbooks as an example of this phenomenon. – Jyrki Lahtonen Jan 30 '24 at 05:36

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