Eisenstein's Criterion

Introduction

Eisenstein's Criterion is a practical tool for determining if a polynomial is irreducible over a field. It is particularly useful for polynomials with integer coefficients and provides a simple test for irreducibility.

Statement

Theorem 11.3 (Eisenstein's Criterion): Let $R$ be a UFD with field of fractions $F$ . Let $f (x) = a_{n} x^{n} + \dots + a_{1} x + a_{0}$ be a polynomial in $R [x]$ . If there exists a prime element $p \in R$ such that:

$p$ divides $a_{i}$ for all $i \in {0, 1, \dots, n - 1}$
$p$ does not divide $a_{n}$
$p^{2}$ does not divide $a_{0}$

then $f (x)$ is irreducible over the field of fractions $F$ .

Proof Sketch

The proof uses contradiction:

Assume $f (x)$ is reducible: $f (x) = g (x) h (x)$ where $\deg (g), \deg (h) < n$
Consider the polynomial modulo $p$ : $\overset{―}{f} (x) = \overset{―}{a_{n}} x^{n}$ in $(R / p R) [x]$
Since $f (x) = g (x) h (x)$ , we have $\overset{―}{f} (x) = \overset{―}{g} (x) \overset{―}{h} (x)$
This implies that $\overset{―}{g} (x) = b x^{k}$ and $\overset{―}{h} (x) = c x^{m}$ for some constants $b, c$
This means that $p$ divides all coefficients of $g (x)$ and $h (x)$ except possibly the leading coefficients
But then $p^{2}$ would divide the constant term of $f (x)$ , contradicting condition 3

Examples

Example 1: Cyclotomic Polynomials

For any prime $p$ , the $p$ -th cyclotomic polynomial $Φ_{p} (x) = \frac{x^{p} - 1}{x - 1} = x^{p - 1} + \dots + x + 1$ is irreducible over $Q$ .

While the criterion does not apply to $Φ_{p} (x)$ directly, it applies to $Φ_{p} (x + 1)$ , which is sufficient to prove the irreducibility of the original polynomial.

Example 2: Polynomials of the Form $x^{n} - p$

The polynomial $x^{n} - p$ is irreducible over $Q$ for any prime $p$ and any positive integer $n$ . This follows directly from Eisenstein's Criterion with the prime $p$ .

Example 3: Polynomials with Prime Constant Term

The polynomial $x^{3} + 2 x^{2} + 4 x + 2$ is irreducible over $Q$ by Eisenstein's Criterion with $p = 2$ .

Applications

Application 1: Constructing Irreducible Polynomials

Eisenstein's Criterion provides a simple way to construct irreducible polynomials over $Q$ .

Application 2: Field Extensions

Irreducible polynomials are crucial for constructing field extensions, and Eisenstein's Criterion helps identify them.

Application 3: Algebraic Number Theory

The criterion is useful in algebraic number theory for studying minimal polynomials of algebraic numbers.

Application 4: Galois Theory

Irreducible polynomials are fundamental to Galois theory, and Eisenstein's Criterion helps identify them.

Limitations

Not Always Applicable

Eisenstein's Criterion is not always applicable. Many irreducible polynomials do not satisfy the conditions of the criterion.

Requires UFD

The criterion requires the coefficient ring to be a UFD, which limits its applicability.

Field of Fractions

The criterion only guarantees irreducibility over the field of fractions, not necessarily over the original ring.