Algebraic number theory MOC

Eisenstein’s criterion

Let be an integral domain and be a polynomial. For a prime ideal , we say is Eisenstein at iff

1. for ;
2. ;
3. .

If is Eisenstein at some prime ideal , then cannot be written as the product of two non-constant polynomials in .¹ alg

Proof

proof

In particular, if is a Unique factorization domain then is also irreducible in , by Gauß’s lemma.

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develop | en | sembr

Footnotes

1. 2009. Algebra: Chapter 0, §V.5.4, p. 288 ↩