Embedding Theorem for the automorphism group of the α-enumeration degrees

It is a theorem of classical Computability Theory that the automorphism group of the enumeration degrees D_e embeds into the automorphism group of the Turing degrees D_T . This follows from the following three statements:
1. D_T embeds to D_e ,
2. D_T is an automorphism base for D_e,
3. D_T is definable in D_e .
The first statement is trivial. The second statement follows from the Selman’s theorem:
A ≤e B ⇐⇒ ∀X ⊆ ω[B ≤e X ⊕ complement(X) implies A ≤e X ⊕ complement(X)].
The third statement follows from the definability of a Kalimullin pair in the α-enumeration degrees D_e and the following theorem:
an enumeration degree is total iff it is trivial or a join of a maximal Kalimullin pair.
Following an analogous pattern, this thesis aims to generalize the results above to the setting of α-Computability theory. The main result of this thesis is Embedding Theorem:
the automorphism group of the α-enumeration degrees D_αe
embeds into the automorphism group of the α-degrees D_α if α is an infinite regular cardinal and assuming the axiom of constructibility V = L. If α is a general admissible ordinal, weaker results are proved involving assumptions on the megaregularity.
In the proof of the definability of D_α in D_αe a helpful concept of α-rational numbers Q_α emerges as a generalization of the rational numbers Q and an analogue of hyperrationals. This is the most valuable theory development of this thesis with many potentially fruitful directions.