Eisenstein's Criterion - Advanced Explanation

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 the lower convex envelope is exactly the single line segment from (0,1) to (n,0), the slope being −1/n.

This tells us that each root of Q has p-adic valuation 1/n and hence that Q is irreducible over the p-adic field (since, for instance, no product of any proper subset of the roots has integer valuation); 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 limit severely the possible choices of p with respect to which the polynomial could have an Eisenstein translate (that is, become Eisenstein after an additive change of variables as in the case of the p-th cyclotomic polynomial).

In fact only primes p ramifying in the extension of ℚ generated by a root of Q have any chance of working. These can be found in terms of the discriminant of Q. For example, in the case x2 + x + 2 given above, the discriminant is −7 so that 7 is the only prime that has a chance of making it satisfy the criterion. Modulo 7, it becomes

(x − 3)2

— a repeated root is inevitable, since the discriminant is 0 mod 7. Therefore the variable shift is actually something predictable.

Again, for the cyclotomic polynomial, it becomes

(x − 1)p − 1 mod p;

the discriminant can be shown to be (up to sign) pp − 2, by linear algebra methods.

More precisely, only totally ramified primes have a chance of being Eisenstein primes for the polynomial. (In quadratic fields, ramification is always total, so the distinction is not seen in the quadratic case like x2 + x + 2 above.) In fact, Eisenstein polynomials are directly linked to totally ramified primes, as follows: if a field extension of the rationals is generated by the root of a polynomial that is Eisenstein at p then p is totally ramified in the extension, and conversely if p is totally ramified in a number field then the field is generated by the root of an Eisenstein polynomial at p.