Krasner's lemma

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Krasner’s lemma is a simple, useful fact in p-adic analysis about how tiny changes to a root affect the field you get by adjoining that root.

Setup: Let K be a complete non-archimedean field (for example, a p-adic field). Let α be an algebraic element over K, and let α2, …, αn be its distinct conjugates over K.

The basic claim is: if β is another element (in the same algebraic closure) that is very close to α, specifically if
|β − α| < min{ |α − α2|, …, |α − αn| },
then the field generated by α is contained in the field generated by β; in symbols, K(α) ⊆ K(β). In words, a small perturbation of α does not create a completely new field—β already carries α inside the field it generates.

A practical view: this means that if you replace α by a root that is very close to α (for example, a root of a nearby linear polynomial), you still generate a field that contains the same algebraic information as α. The original, simplest form of Krasner’s lemma corresponds to the case where the helper polynomial has degree 1.

General idea: there is a broader version that works for monic polynomials of any degree over a Henselian field. It compares two polynomials whose roots are grouped in two nonempty sets and says that, under suitable closeness conditions, the field generated by one set of coefficients is contained in the field generated by the other. This general picture extends the basic idea: closeness of roots enforces a containment of the corresponding generated fields.