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### Completion of infinite degree extension of perfectoid fields is perfectoid?

I'm assuming you mean infinite algebraic extensions, as otherwise there is no standard way of completing them. Let $K$ be a perfectoid field, let $L$ be an infinite algebraic extension. Then $L$ ...
• 23.7k

### Galois groups vs. fundamental groups

I found this introduction by David Corwin very accessible and well explained: https://math.berkeley.edu/~dcorwin/files/etale.pdf.
• 3,804
There would be no consequences, for two reasons: As Timothy Chow points out, if we define $\overline{\mathbb Q}$ as the set of complex numbers that are roots of a nonzero polynomial with rational ...
In the absence of the axiom of choice, it is still possible to define the "usual" algebraic closure of $\mathbb{Q}$ because you can just explicitly enumerate all polynomials with integer ...