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Eisenstein's Irreducibility Criterion — Definition, Formula & Examples

Eisenstein's Irreducibility Criterion is a test that uses a single prime number to prove a polynomial with integer coefficients cannot be factored into polynomials of lower degree over the rationals.

Let f(x)=anxn+an−1xn−1+⋯+a1x+a0f(x) = a_n x^n + a_{n-1}x^{n-1} + \cdots + a_1 x + a_0 be a polynomial in Z[x]\mathbb{Z}[x] with deg⁡f≥1\deg f \geq 1. If there exists a prime pp such that (1) p∤anp \nmid a_n, (2) p∣aip \mid a_i for all 0≤i≤n−10 \leq i \leq n-1, and (3) p2∤a0p^2 \nmid a_0, then f(x)f(x) is irreducible over Q\mathbb{Q}.

Key Formula

f(x)=anxn+an−1xn−1+⋯+a1x+a0f(x) = a_n x^n + a_{n-1}x^{n-1} + \cdots + a_1 x + a_0
Where:
  • ana_n = Leading coefficient, not divisible by the chosen prime p
  • a0,…,an−1a_0, \ldots, a_{n-1} = Remaining coefficients, each divisible by p
  • a0a_0 = Constant term, not divisible by p²
  • pp = A prime number satisfying all three conditions

How It Works

To apply the criterion, look for a single prime pp that divides every coefficient except the leading one, and whose square does not divide the constant term. If such a prime exists, the polynomial is irreducible over Q\mathbb{Q} — no further work is needed. The criterion is sufficient but not necessary: many irreducible polynomials fail it for every prime. A common strategy is to apply a substitution like x↦x+cx \mapsto x + c to transform the polynomial into one where the criterion does apply.

Worked Example

Problem: Prove that f(x)=x4+6x3+9x2+12x+3f(x) = x^4 + 6x^3 + 9x^2 + 12x + 3 is irreducible over Q\mathbb{Q}.
Choose a prime: Try p=3p = 3. The coefficients are 1,6,9,12,31, 6, 9, 12, 3.
Check condition 1: The leading coefficient is a4=1a_4 = 1, and 3∤13 \nmid 1. Condition satisfied.
Check condition 2: The remaining coefficients are 6,9,12,36, 9, 12, 3. Each is divisible by 33. Condition satisfied.
3∣6,3∣9,3∣12,3∣33 \mid 6,\quad 3 \mid 9,\quad 3 \mid 12,\quad 3 \mid 3
Check condition 3: The constant term is a0=3a_0 = 3, and 32=9∤33^2 = 9 \nmid 3. Condition satisfied.
Answer: All three conditions hold for p=3p = 3, so f(x)f(x) is irreducible over Q\mathbb{Q} by Eisenstein's criterion.

Why It Matters

Eisenstein's criterion appears throughout abstract algebra and number theory courses. It provides the standard proof that the cyclotomic polynomial Φp(x)=xp−1+xp−2+⋯+x+1\Phi_p(x) = x^{p-1} + x^{p-2} + \cdots + x + 1 is irreducible for any prime pp (after substituting x↦x+1x \mapsto x+1). It is also a key tool in constructing field extensions of specified degree.

Common Mistakes

Mistake: Forgetting to check that p2p^2 does not divide the constant term.
Correction: All three conditions are required. For example, x2+4x+4=(x+2)2x^2 + 4x + 4 = (x+2)^2 is reducible even though p=2p = 2 divides every non-leading coefficient, because 22∣42^2 \mid 4 and condition 3 fails.