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Prime numbers, diophantine equations, diophantine approximations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions

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Is it known that certain extensions of the integers are UFDs?

For an integer $n$, let $\omega=\sqrt n$ unless $n=\equiv 1 \pmod 4$ and in that case let $\omega = (1+\sqrt n)/2$. Then the ring $Z[\omega]$ is well known to be integrally closed. For $p$ a prime, …