Solvably Closed Field

A solvably closed field is a field having no nontrivial finite solvable Galois extension. Equivalently, it has no nontrivial abelian algebraic extension. Algebraic closures are examples, and the maximal prosolvable extension of a number field is another example. Solvable closure is weaker than algebraic closure, since extensions with nonsolvable Galois groups can remain.

Galois groups of solvably closed Galois extensions of number fields are the objects in Uchida's homomorphism conjecture.