Eisenstein's Criterion: Irreducibility Test Explained

Eisenstein's criterion proves a polynomial with integer coefficients is irreducible over ℚ — unfactorable into lower-degree rational polynomials. For f(x) = a_n x^n + … + a_0, if some prime p divides every coefficient except the leading one, does not divide the leading coefficient, and p² does not divide the constant term, then f is irreducible over ℚ.

Irreducibility is usually hard — trying all factorizations is hopeless beyond low degrees. Eisenstein settles it in one glance, and it is the standard tool for building field extensions: an irreducible degree-n polynomial yields an extension of degree n.

Key Concepts

• Irreducible over ℚ: not a product of two non-constant rational polynomials.
• Gauss's lemma: factorization over ℚ equals factorization over ℤ (up to ±1) — integer divisibility controls it.
• The three conditions: p ∤ a_n; p | a_i for i < n; p² ∤ a_0.
• Proof idea: if f = gh in ℤ[x], p² ∤ a_0 forces p to divide exactly one constant term, hence all coefficients of one factor — contradicting p ∤ a_n unless a factor is constant.
• Shifts help: if f(x) fails, f(x + c) might pass; shifts preserve irreducibility.

Formulas

• The criterion: f(x) = Σ a_i x^i, a_i ∈ ℤ: if some prime p has p ∤ a_n, p | a_0…a_{n−1}, p² ∤ a_0, then f is irreducible over ℚ. (Divisibility of integers; no physical units.)
• Classic family: x^n − p is irreducible for every prime p, n ≥ 1.
• Limit: sufficient but not necessary — x² + 1 is irreducible over ℚ yet fails for every prime.

Worked Example

Prove f(x) = x³ + 2x + 2 is irreducible over ℚ, using p = 2. Coefficients: 1, 0, 2, 2. Condition 1: 2 ∤ 1. Condition 2: 2 | 0, 2 | 2, 2 | 2. Condition 3: 4 ∤ 2. All hold — f is irreducible over ℚ. Sanity check via the rational root theorem: possible roots ±1, ±2, but f(1) = 5, f(−1) = −1, f(2) = 14, f(−2) = −10 — no zeros, consistent (a cubic that factors must have a linear factor, hence a root).

Common Misconceptions

• "If no prime works, the polynomial is reducible." The test is one-directional: x² + 1 and x^4 + 1 are irreducible over ℚ but fail Eisenstein for every prime (try a shift first).
• "p² ∤ a_0 is a minor technicality." It is essential: x² + 4x + 4 = (x + 2)² satisfies the first two conditions with p = 2 but is reducible — and indeed 4 | 4.
• "Irreducible over ℚ means no roots anywhere." It means no factorization over ℚ; x³ + 2x + 2 has one real root (approximately −0.77) and factors over ℝ — irreducibility is always relative to the coefficient field.

Real-World Applications

• Field extensions: irreducible polynomials build extension fields — ℚ(∛2) comes from x³ − 2, irreducible by Eisenstein with p = 2.
• Galois theory: irreducibility of the minimal polynomial controls the size and structure of Galois groups.
• Finite fields: constructing GF(p^n) needs an irreducible degree-n polynomial over ℤ/pℤ; Eisenstein-style criteria guide the search.
• Algebraic number theory: integral bases of number fields are built from Eisenstein polynomials.

Eisenstein's criterion turns three quick divisibility checks into a proof of irreducibility — a prime dividing all but the leading coefficient, with its square sparing the constant term, guarantees the polynomial cannot split over ℚ.