A perfectoid field K is defined when it satisfies three conditions: ① it is complete, ② its value group is dense in ℝ, and ③ the p-th power map on O_K/pO_K is surjective. In the case of positive characteristic, this is equivalent to saying it is a complete perfect field (since O_K/pO_K = O_K, and moreover the density of the value group follows immediately from O_K^p = O_K (discrete valuations are characterized by having an element x with positive minimal valuation, so if there exists y ∈ K such that x = y^p, this contradicts minimality)).

There exists a tilting correspondence that connects perfectoid fields of positive characteristic with perfectoid fields of characteristic 0 (where the operations of taking ♭ and taking ♯ are inverse maps to each other). By combining this with results obtained using Galois cohomology that establishes an isomorphism between Gal(K₁^♭/K^♭) and Gal(K₁/K), one can show that finite extensions of perfectoid fields are perfectoid fields (see Galois representation and (φ,Γ)-modules p.74). By taking the direct limit (union, since we are dealing with fields) of these, one can verify that even in the case of algebraic extensions, algebraic extensions of perfectoid fields are perfectoid.

The field in this example is the completion of the union of two representative perfectoid fields (with the same value group). Since the union over one of the fields is algebraic, it is perfectoid.

To verify this directly, for condition ③, it would be good to write the residue ring of the ring of integers as a direct union in the case of finite extensions, thereby reducing to the finite case.

Note that trying to compute the ring of integers directly would be difficult since the ramification index and other quantities would be unknown.