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Last year, in a talk of Michel Waldschmidt's, I remember hearing a statement along the lines of the title of this question:

The Galois group of $\pi$ is $\mathbb{Z}$.

In what sense/framework is this true? What was meant exactly - and can this notion be made precise?

I imagine that this would draw on the fact that the $\sin x$ is an 'infinite polynomial' over $\mathbb{Q}$ whose roots are the integer multiples of $\pi$. However, if this is the case, then how can we possibly view this function as a sort of generalized 'minimal polynomial' while we have factorizations like $\sin x = 2 \cos (x/2) \cdot \sin (x/2)$?

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    Related: https://math.stackexchange.com/questions/19609692017-01-08

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