No, because you cannot apply Eisenstein here: ##3 \nmid 16 = a_0##.Mr Davis 97 said: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?
The Eisenstein Criterion, also known as the Eisenstein's irreducibility criterion, is a mathematical test used to determine whether a polynomial with integer coefficients is irreducible over the rational numbers.
The Eisenstein Criterion states that if a polynomial has integer coefficients and can be written as a sum of polynomials, where all but one of the terms has a coefficient divisible by a prime p, and the remaining term has a coefficient not divisible by p, then the polynomial is irreducible over the rational numbers.
The Eisenstein Criterion is significant because it provides a quick and effective way to determine the irreducibility of a polynomial without having to use more complex methods. This criterion is particularly useful in algebraic number theory and algebraic geometry.
The Eisenstein Criterion should be used when trying to determine the irreducibility of a polynomial with integer coefficients over the rational numbers. It is also useful in determining whether a polynomial is a prime element in a polynomial ring.
Yes, there are some limitations to the Eisenstein Criterion. It can only be used for polynomials with integer coefficients, and the prime number p used in the test must be chosen carefully. Additionally, the Eisenstein Criterion does not always guarantee that a polynomial is irreducible, but it can provide a quick way to eliminate certain cases.