In classical algebra, the prime fields are Q and for every prime number p the finite field F_p. In higher algebra, one has for every prime number p an additional sequence of prime fields K(p,n), n a natural number, which in some sense interpolates between Q and F_p. Associated with these prime fields one has corresponding localization and completion functors. An interesting question, raised by Waldhausen and Ausoni—Rognes, is how these functors interact with algebraic K-theory. In the talk I will first give an introduction and discuss a purity result for algebraic K-theory with respect to these completion functors. This is based on joint work with Markus Land, Akhil Mathew, and Lennart Meier and on closely related work of Dustin Clausen, Akhil Mathew, Niko Naumann, and Justin Noel.