Unramified algebraic extensions of commutative Banach algebras

Extensions of a commutative Banach algebra $A$ by means of roots of polynomials over $A$ with invertible discriminant are investigated. In the case when $A$ has no nontrivial idempotent, for each such polynomial $f$ a Banach algebra $A_f$, which plays the role of a minimal splitting algebra, is constructed. Unramified radical extensions of $A$ are defined, and the question of the solvability of algebraic equations over $A$ in unramified radicals is investigated.
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