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We answer an open question in the theory of degrees of infinite sequenceswith respect to transducibility by finite-state transducers. An initial studyof this partial order of degrees was carried out in (Endrullis, Hendriks, Klop,2011), but many basic questions remain unanswered. One of the central questionsconcerns the existence of atom degrees, other than the degree of the `identitysequence' 1 0^0 1 0^1 1 0^2 1 0^3 .... A degree is called an `atom' if below itthere is only the bottom degree 0, which consists of the ultimately periodicsequences. We show that also the degree of the `squares sequence' 1 0^0 1 0^1 10^4 1 0^9 1 0^{16} ... is an atom. As the main tool for this result wecharacterise the transducts of `spiralling' sequences and their degrees. We usethis to show that every transduct of a `polynomial sequence' either is in 0 orcan be transduced back to a polynomial sequence for a polynomial of the sameorder.

The Degree of Squares is an Atom (Extended Version)