This expository paper is a companion to \cite{Rav:gjmcyc}, in which we discuss cyclotomic spectra. Both papers are intended to shed light on the recent resolution of the telescope conjecture by Robert Burklund, Jeremy Hahn, Ishan Levy and Tomer Schlank (hereafter referred to as BHLS) in \cite{BHLS}. Their proof involves both cyclotomic spectra, the subject of \cite{Rav:gjmcyc}, and cyclotomic extensions of spectra, the subject of this paper. Higher cyclotomic extensions of commutative ring spectra are analogous to Galois extensions of $p$-adic number fields (or rings of integers thereof) obtained by adjoining roots of unity.

75 pages