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The Galois Group of $\mathbb{R}$ as an extension of $\mathbb{Q}$ is trivial. That’s because any field automorphism of $\mathbb{R}$ is order preserving, so since $\mathbb{Q}$ is dense in $\mathbb{R}$, it follows that the only field automorphism of $\mathbb{R}$ that fixes $\mathbb{Q}$ is the identity.

But I don’t think $\mathbb{Q}$ is dense in the field of hyperreal numbers. So my question is, what is the Galois group of the field of hyperreal numbers as an extension of $\mathbb{Q}$?

I expect it to be a pretty large group, maybe something that involves a Hamel basis of $\mathbb{R}$ over $\mathbb{Q}$.

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