Do there exist non-trivial absolutely normal field extensions? (Q689821)

Let \(L/K\) be a -- possibly transcendental -- extension of (commutative) fields. We call \(L/K\) absolutely normal if, for any intermediate field \(E\) of \(L/K\), the relative algebraic closure of \(E\) in \(L\) is a normal extension of \(E\) in the usual sense. Trivial examples of such extensions are, besides normal algebraic extensions, algebraically closed extensions of arbitrary transcendency degree. So far, we do not know if there exist any non-trivial examples. We can show, however, that if \(L/K\) is an absolutely normal transcendental extension, then \([L:K(T)]=\infty\), for any transcendency basis \(T\) of \(L/K\).