Eisenstein's criterion

Eisenstein's criterion, in mathematics, is a method of determining whether a polynomial is irreducible.

Suppose we have a polynomial with integer coefficients.

$f(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\ldots +a_{1}x+a_{0}$

If there exists a prime number p, and p divides all a_i except a_n, and p² does not divide a₀, f(x) is irreducible.

Basic proof

Consider f(x) as a polynomial modulo p; that is, reduce the coefficients to the field Z/pZ. There it becomes c.xⁿ for a non-zero constant c. Because such polynomials factorise uniquely, any factorisation of f mod p must be into monomials. Now if f were not irreducible as an integer polynomial, we could write it as g.h, and f mod p as the product of g mod p and h mod p. These latter must be monomials, as has just been said, meaning that we have g mod p is d.x^k and h mod p is e.x^n-k where c = d.e.

Now we see that the conditions given on g mod p and h mod p mean that p² will divide a₀, a contradiction to the assumption. In fact a₀ will be g(0).h(0) and p divides both factors, from what was said above.

Advanced explanation

Applying the theory of the Newton polygon for the p-adic number field, for an Eisenstein polynomial, we are supposed to take the lower convex envelope of the points (0,1), (1, v₁), (2, v₂), ..., (n-1, v_n-1), (n,0), where v_i is the p-adic valuation of a_i (i.e. the highest power of p dividing it). Now the data we are given on the v_i for 0 < i < n, namely that they are at least one, is just what we need to conclude that lower convex envelope is exactly the single line segment from (0,1) to (n,0), the slope being - 1/n.

From the general theory we know that p then ramifies completely in the extension of the p-adic numbers generated by a root of f. For that reason f is irreducible over the p-adic field; and a fortiori over the rational number field.

This argument is much more complicated than the direct argument by reduction mod p. It does however allow one to see, in terms of algebraic number theory, how frequently Eisenstein's criterion might apply, after some change of variable; and so, bound the possible choice of p. In fact only primes p ramifying in the extension of Q generated by a root of f have any chance of working. These can be found in terms of the discriminant of f.