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Results tagged with galois-theory
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user 1384
Galois theory, named after Évariste Galois, provides a connection between field theory and group theory. Using Galois theory, certain problems in field theory can be reduced to group theory, which is, in some sense, simpler and better understood.
14
votes
Accepted
Totally ramified subextension in a finite extension of $\mathbf{Q}_p$
This is not a complete answer, but perhaps it's a roadmap to a counterexample.
My strategy is to consider some non-Galois $K/\mathbf{Q}_p$ for which the result is true, and let's make some deductions …
- 41.4k
4
votes
Accepted
Galois group of an L-function
Let $M$ be the set of finite products of Dirichlet $L$-functions. These surely form a class of $L$-functions as in the question. Now take some prime $p$ congruent to 1 mod 4 and let $\chi$ be one of t …
- 41.4k
27
votes
"Understanding" $\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$
I am pretty sure that when different number theorists say "one of the main goals of number theory is to understand Gal(Q-bar/Q)" they may well mean different things.
One example of what someone might …
- 41.4k
13
votes
$A_5$-extension of number fields unramified everywhere
Oh, I know how I would try and build examples. First I would write down a random $A_5$ extension $K$ of $\mathbf{Q}$, ramified at some primes (in fact I would look in a table, e.g. in Buhler's thesis …
- 41.4k