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Results tagged with absolute-galois-group
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user 18060
3
votes
Accepted
Galois action on ultrapowers
This is quite trivial but not enough for a comment: $F^G$ is not always $K^\mathbb N/\mathcal U$.
Let $x_1,x_2,\dots$ be elements of $K$ such that no product of finitely many of them is a perfect squ …
- 148k
3
votes
Accepted
non-continuous inverse Galois problem
Q1: No.
Suppose there were such a subgroup. Then there would certainly be a $K$ such that $G/K$ is $\mathbb Z$. $K$ would have to contain the commutator subgroup. The quotient of $G$ by the commutato …
- 148k
17
votes
Absolute Galois group, number theory and the Axiom of Choice
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 coe …
- 148k
8
votes
Accepted
Galois action on automorphisms of a curve
$K$-defined would mean that the subgroup itself is defined over $K$, not the elements.
Formally, this means that there are equations over $K$ which are satisfied by the coefficients of polynomials def …
- 148k
17
votes
Subgroups of GL(2,q)
We can generalize Ralph's answer to find the abelian groups that are $p$-groups for odd $p$ that cannot be subgroups of $GL_2$.
if $p|(q^2-1)(q^2-q)$ then exactly one of $p|(q+1)$, $p|(q-1)$ and $p|q …
- 148k