Determine whether the polynomial $x^2-12$ in $\mathbb Z[x]$ satisfies an Eisenstein criterion for irreducibility over $\mathbb Q$

Question

Determine whether the polynomial $x^2-12$ in $\mathbb Z[x]$ satisfies an Eisenstein criterion for irreducibility over $\mathbb Q$.

So If I understand correctly, I start with $x^2=12$ and then you want to find a $p$ such that $12=0\pmod p$ which we can use $2$ for our $p$ and we are looking for all $n < 2$ to not equal $0 \bmod 2$, but for all $n > 2$ to equal $0 \bmod 2$?

So we would just do:

x=1 1 is not congruent to 0 mod 2
x=2 0 is congruent to 0 mod 2
x=-1 1 which is not congruent to 0 mod 2,
however, x=-2 is congruent to 0 mod 2, therefore the polynomial does not satisfy the Eisenstein criterion.

I am just trying to make sure that I am doing this correctly. Also, if this is correct then another possible p would be 3, so would you just do a chart for both of them and start plugging in numbers? How do you know which p to use if not?

Answer 1

So as you claim $a_0\neq 0\pmod{p^2}$. So, you cannot choose $p$ to be $2$ as $2^2|12$.

Taking any other prime that divides $12$ would do the trick. And there is only one such prime, $3$

Answer 2

Eisenstein's Criterion for $\mathbb{Z}[x]$: Let $p \in \mathbb{Z}$ be a prime, and let $f(x) = x^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0 \in \mathbb{Z}[x], n \geq 1$. Suppose $p$ divides $a_i$ for all $i \in \{0, 1, \cdots , n-1\}$ but that $p^2$ does not divide $a_0$. Then $f(x)$ is irreducible in both $\mathbb{Z}[x]$ and $\mathbb{Q}[x]$.

Let $g(x) = x^2 - 12$. Define $f(x) := g(x +3) = x^2 + 6x - 3$. Choose $p = 3$. Then by Eisenstein's Criterion, $f(x)$ is irreducible in both $\mathbb{Z}[x]$ and $\mathbb{Q}[x]$. This implies that $g(x)$ is also irreducible in both $\mathbb{Z}[x]$ and $\mathbb{Q}[x]$, beacause any factorization of $g(x)$ will also give a factorization of $f(x)$ after replacing $x$ by $x + 3$ in each factor.

For some polynomials we can't use Eisenstein's Criterion directly. We need to change the polynomial a little so that we can use it. $x^2 - 12$ is such an example. You can also try for the following polynomials: $x^4 + 1, x^2 + x + 1, x^8 - 1, x^6 - 1$.

EDIT: We can use Eisenstein's criterion for the polynomial $x^2 - 12$ directly by taking $p = 3$. We don't need to go for the change of variables. Thanks to Gerry Myerson for pointing it. I missed it completely.