Residue field of completion of a field finite type over the prime field

In Theory of p-adic Galois Representations by Fontaine and Yi Ouyang, there is a proof of Grothendieck l-adic monodromy theorem for general local fields:

Here a local field is a complete discrete valuation field with perfect residue field. But I could not understand why $k_1$ is finite type over the prime field. Is it trivially true? What is the original proof?