Proving irreducible polynomial using Eisenstein's criteria.

Question

How can I prove that using the Eisenstein's criteria, that the polynomial : $x^3 -4x +2$ is irreducible polynomial for prime $2$?

I am new to this topic, and while attempting would try to get the logic of the criteria too.

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My attempt is : for $x=2$, the value of $a_0 = 2, a_1= -8, a_2 = 0, a_3 =8$.

To apply Eisenstein's criterion there should be a prime $p$ such that

(i) $p\mid a_0, a_1,..., a_{n-1} $ => passes as $2 \mid 2, -8, 0$

(ii) $p \nmid a_n$ => fails as $2 \mid 8$, this also means that the highest term (let, of the power $n$) must be multiplied by a suitable integer multiplier (let, $m$), so that $p \nmid m.p^n$. I am not aware of its algebraic proof, or even its significance, or ramifications; but it is just an observation. It also means that for monic polynomials it must always fail, as $p \mid p^n => p \mid a_n $.

(iii) $p^2 \nmid a_0$ => passes as $4 \nmid 2$, this also means that $c$ term must be non-zero and composite for this condition to be true.

As given above, the second criteria has failed, so where is the flaw in my attempt?

I have another question, how will one test for any prime fitting in for test, i.e. there must be some restrictions on the choice of primes to be tested. And how it is possible that there is only one value enough for pass/fail.

Answer 1

Because $0$, $4$ and $2$ they are divisible by $2$, but the last coefficient $2$ is not divisible by $4$.

For irreducibility it's enough to use the Eisenstein's criteria for one prime number.

Answer 2

Your flaw is in the $a_{i}$'s. These are the coefficients of the polynomial, i.e. in your case

$$p(x)=a_{3}x^{3}+a_{2}x^{2}+a_{1}x+a_{0}$$

$$\Longrightarrow a_{3}=1,\:a_{2}=0,\:a_{1}=-4,\:a_{0}=2$$

and not the value of each monomial evaluated at a specific $x$.