Eisenstein's criterion

Eisenstein's criterion is a sufficient test for a polynomial with integer coefficients to be irreducible over the rationals: one prime must divide every nonleading coefficient but not the constant term's square.

Polynomial ring: A ring whose elements are polynomials with coefficients in a specified ring. The criterion applies to polynomials over the integers and concludes irreducibility over the rationals.
Eisenstein polynomial: An integer polynomial whose coefficients satisfy Eisenstein's divisibility conditions for some prime. This is the direct class of examples to which the criterion applies.
Rational root theorem: A theorem restricting the possible rational roots of a polynomial with integer coefficients. It can establish reducibility through a root but generally does not prove irreducibility by itself.
Gottlob Eisenstein: A German mathematician whose work included major contributions to number theory and algebra. His name is attached to the irreducibility criterion.
Irreducible polynomial: A nonconstant polynomial that cannot be expressed as a product of two nonconstant polynomials over a specified coefficient field or ring. This is the property Eisenstein's test certifies over the rational numbers.
Cyclotomic polynomial: A polynomial whose roots are the primitive roots of unity of a specified order. A shifted cyclotomic polynomial provides a standard use of Eisenstein's criterion in proving irreducibility.
Perron's criterion: A sufficient test for polynomial irreducibility based on one coefficient dominating the sum of the others. It offers a different coefficient-based route when Eisenstein's divisibility conditions fail.
Ferdinand Gotthold Max Eisenstein: The full name commonly associated with the nineteenth-century mathematician Gottlob Eisenstein. This name variant identifies the same mathematician behind the criterion's eponym.
Gauss's lemma: A result stating that a primitive integer polynomial factors over the rationals only if it factors over the integers. It transfers the rational factorization question to integer factors, where prime divisibility can constrain them.
xⁿ − 2: The polynomial obtained by subtracting 2 from the nth power of the variable x. For every positive n, the prime 2 meets the criterion's conditions.

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Gotthold EisensteinBroader topic: This named criterion gives a concise example of Eisenstein’s algebraic ingenuity.