Abstract This paper is devoted to irreducible radical field extensions, i.e., extensions that can be obtained by adjunction of roots of irreducible binomials. We find a criterion for cyclotomic extensions to be irreducible radical. This criterion developes the Gauss–Wantzel theorem about constructible polygons. We also prove that any normal solvable extension of some field K is irreducible radical until K has all roots of unity. This generalizes Abel’s theorem, which fills the gap in the uncomplete Ruffini proof of the impossibility theorem for the general equation of degree five or higher. Finally, we prove that the root set of any irreducible solvable polynomial coincides with the value set of some radical formula using irreducible radical extensions.
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