A theorem of Artin and Schreier asserts that (essentially) these are all the possibilities for finite absolute Galois groups.
Then there exists a unique isomorphism of the algebraic closures, b: Kalg → Lalg, that induces a.
This generalizes an earlier work of Jürgen Neukirch and Koji Uchida on number fields.
Michael Fried and Helmut Völklein applied algebraic topology and complex analysis to establish Roquette's conjecture in characteristic zero.
Later Pop proved the Theorem for arbitrary characteristic by developing "rigid patching".