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   We consider first-order definability and decidability questions over rings of integers of algebraic extensions of $$\Q$$, paying attention to the uniformity of definitions. The uniformity follows from the simplicity of our first-order definition of $$\Z$$. Namely, we prove that for a large collection of algebraic extensions $$K/\Q$$, $$ \{x \in \oo_K : \text{$$\forall \e \in \oo_K^\times \;\exists \delta \in \oo_K^\times$ such that $$\delta-1 \equiv (\e-1)x \pmod{(\e-1)^2}$$}\} = \Z $$ where $$\oo_K$$ denotes the ring of integers of $$K$$. One of the corollaries of our results is undecidability of the field of constructible numbers, a question posed by Tarski in 1948. \end{abstract} 

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5. Generalized $$\mathbb Z$$-homotopy fixed points of $$C_n$$ spectra with applications to norms of $$MU_{\mathbb R}$$

   Hill, Michael A; Zeng, Mingcong (January 2020, New York journal of mathematics)

   null (Ed.)

   We introduce a computationally tractable way to describe the $$\mathbb Z$$-homotopy fixed points of a $$C_{n}$$-spectrum $$E$$, producing a genuine $$C_{n}$$ spectrum $$E^{hn\mathbb Z}$$ whose fixed and homotopy fixed points agree and are the $$\mathbb Z$$-homotopy fixed points of $$E$$. These form the bottom piece of a contravariant functor from the divisor poset of $$n$$ to genuine $$C_{n}$$-spectra, and when $$E$$ is an $$N_{\infty}$$-ring spectrum, this functor lifts to a functor of $$N_{\infty}$$-ring spectra. For spectra like the Real Johnson--Wilson theories or the norms of Real bordism, the slice spectral sequence provides a way to easily compute the $RO(G)$-graded homotopy groups of the spectrum $$E^{hn\mathbb Z}$$, giving the homotopy groups of the $$\mathbb Z$$-homotopy fixed points. For the more general spectra in the contravariant functor, the slice spectral sequences interpolate between the one for the norm of Real bordism and the especially simple $$\mathbb Z$$-homotopy fixed point case, giving us a family of new tools to simplify slice computations. 

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