Some More Rigid Real Closed Fields

We extend a construction of Marker and Steinhorn for producing countable non-archimedean real closed fields which have no non-trivial automorphisms. The original construction produces such a field of transcendence degree two over the field of real algebraic numbers. We show how to use the same method to produce examples of all finite transcendence degrees and of countably infinite transcendence degree over the real algebraic numbers, answering a question of Marker and Steinhorn.