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3ds max 2011 serial number 32 bit keygen for windows 7. 3ds Max 2011 serial number 32 bit keygen for windows 7 is a powerful 3D software for rendering and modeling computer graphics 3d professional is a 3d max free download,. 3ds max 2011 serial number 32 bit keygen for windows 7.Q:

Is $\Bbb P^\times_\Bbb Q$ a field?

I am wondering if $\Bbb P^\times_\Bbb Q$ is a field?
I know that the rationals are a field and $\Bbb P^1_\Bbb Q$ is a field. The first inclusion is a subgroup, but the second inclusion is not easy to prove.

A:

$ewcommand{\Q}{\Bbb Q}$It is a group, but not a field (or a vector space over $\Q$). The proof requires a trick you might learn in algebraic topology: let $f:P^1_{\Q}\to P^1_{\Q}$ be a function. We say $f$ is defined $r$-locally in $x\in P^1_{\Q}$ if, for all $t\in\Q$, $f(x+t)-f(x)=t^r$. We say $f$ is an injection if it is an injection in all points $x+t$, where $x$ ranges over points with $r$-local image.
Suppose $f:P^1_{\Q}\to P^1_{\Q}$ is a function. The first fundamental group of $P^1_{\Q}$ is $\pi_1(P^1_{\Q})=\Q$ (with the homotopy class of a point as a generator), and the image of the covering map $\pi_1:P^1_{\Q}\to\Q$ is $\pi_1(\Q)=\Q$ itself (again, the homotopy class of a point is a generator). Thus the image of $f$ is contained in the group generated by the image of the covering

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