In this work, we start with a new method to adjoin roots to E_2 ring spectra. We prove that this process is logarithmic THH \'etale and that it results in interesting splittings in algebraic K-theory. For instance, we obtain that T(n+1)-locally, the algebraic K-theory of the nth Morava E-theory contains the algebraic K-theory of the truncated Brown-Peterson spectrum with coefficients as a summand. In particular, this provides a proof of the fact that Morava E-theories satisfy the redshift conjecture. This is a joint work with Tasos Moulinos and Christian Ausoni.

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