Eisenstein Criterion: Irreducibility Test

[Mr Davis 97]

Homework Statement

Determine the irreducibility $x^2 - 12$

Homework Equations

The Attempt at a Solution

By using $p=3$ we see that $x^2 - 12$ is irreducible, because 3 does not divide 1, and 9 does not divide 12. That's easy enough.
But what if I have $x^2 - 16$? Obviously this is factorable, but using p=3, doesn't the Eisenstein criterion tell us that it is irreducible?

 

[fresh_42]

Homework Statement

Determine the irreducibility $x^2 - 12$

Homework Equations

The Attempt at a Solution

By using $p=3$ we see that $x^2 - 12$ is irreducible, because 3 does not divide 1, and 9 does not divide 12. That's easy enough.
But what if I have $x^2 - 16$? Obviously this is factorable, but using p=3, doesn't the Eisenstein criterion tell us that it is irreducible?

No, because you cannot apply Eisenstein here: $3 \nmid 16 = a_0$.