# “The Galois group of $\pi$ is $\mathbb{Z}$”

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 $\sin x = 2 \cos (x/2) \cdot \sin (x/2)$?

• You could consider the Galois group of $\mathbb{Q}(\pi)$ over $\mathbb{Q}$. Since $\mathbb{Q}(\pi)\cong \mathbb{Q}(x)$, this would be equivalent to asking for the set of all $\mathbb{Q}$-automorphism of the function field $\mathbb{Q}(x)$. But then you would get the group of invertible $2\times 2$ matrices with coefficients in $\mathbb{Q}$, rather than $\mathbb{Z}$. E.g., here. – Arturo Magidin Dec 29 '11 at 18:27
• My guess is that $\pi$ is (related to) the period attached to the motive attached to the multiplicative group scheme $\mathbb{G}_m$, which over the complex numbers is $\mathbb{C}^{\times}$, which has fundamental group $\mathbb{Z}$, so should have etale fundamental group something like $\mathbb{Z}$ or its profinite completion, and etale fundamental groups generalize Galois groups. (Actually from mathoverflow.net/questions/15153/periods-and-l-values it looks like the right period is $2\pi i$.) – Qiaochu Yuan Dec 29 '11 at 18:28
• @Arturo: don't you mean $\text{PGL}_2(\mathbb{Q})$? – Qiaochu Yuan Dec 29 '11 at 18:33
• Crossposted to MO – t.b. Dec 30 '11 at 2:53
• Another explanation (for lesser mortals than @Qiaochu) is that $2\pi i$ is the period of $\exp(z)$ defined over $\Bbb C$ so that the Riemannsurface $w = \log(z)$ forms an infinite-sheeted covering of the complex plane ramified at $z = 0$ and pasted togather at the negative real axis. For a point $x$ traversing a loop around the ramification point, $\log(x)$ winds upwards to $\log(x) + 2\pi i$ in the sheet above instead of coming back. This indicates that the monodromy of the cover is infinite cyclic, i.e., $\Bbb Z$, and monodrmies generalize Galois group. – Balarka Sen Jul 9 '14 at 14:09