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Results tagged with galois-theory
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user 4213
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.
20
votes
$A_5$-extension of number fields unramified everywhere
Here's the standard example. I found it in Lang's Algebraic Number Theory
where he attributes it to Artin. Let $K$ be the splitting field of $X^5-X+1$
over $\mathbb{Q}$. Then $K$ has Galois group $S_5 …
- 41.4k
14
votes
Why is every quadratic subfield of a Galois extension of the rationals with the quaternions ...
To expand slightly my brief comment. Regard $L$ as a subfield
of $\mathbb{C}$. Let $L'=\mathbb{R}\cap L$. Then the index
$|L:L'|=1$ or $2$. In the former case all quadratic subfields of
$L$ are real. …
- 20.8k
29
votes
Accepted
When is sin(r \pi) expressible in radicals for r rational?
As $\cos x=\pm\sqrt{1-\sin^2 x}$ and $e^{ix}=\cos x +i\sin x$, and
$\sin x=(e^{ix}-1/e^{ix})/2i$ then $\sin x$ is in a radical extension of $\mathbb{Q}$
iff $e^{ix}$ is. For rational $r$ with denomina …
- 20.8k
10
votes
Fields with trivial automorphism group
There are examples involving the $p$-adics: for instance $\mathbb{Q}_p$
itself has trivial automorphism group. Indeed as $\mathbb{Q}(i)$
embeds in $\mathbb{Q}_p$ when $p\equiv1$ (mod 4) then $\mathbb{ …
- 20.8k
5
votes
Is there a notion of Galois extension for Z / p^2?
There's the notion of Galois ring. Let $K$ be the degree $m$
unramified extension of $\mathbb{Q}_p$
and let $\mathcal{O}_K$ be its ring of integers. Then the quotient
$R=\mathcal{O}_K/p^n\mathcal{O}_ …
- 20.8k
20
votes
Accepted
Galois group of a product of irreducible polynomials
The Galois group of $P$ will be a subdirect product of
the $G_i$, that is a subgroup of $G_1\times\cdots\times G_k$
projecting surjectively onto each of the $G_i$.
- 20.8k
1
vote
What are Mean Values of Ideal Densities in Galois Extensions?
This may be a reference to the the Davenport-Heilbronn theorem
on the distribution of discriminants of number fields.
See
math.stanford.edu/~fthorne/davenport-heilbronn.pdf
for a nice exposition. Str …
- 20.8k
9
votes
Accepted
A special integral polynomial
An easy way to ensure that a polynomial $g$ of degree $m$ over $\mathbf{Z}$ has
Galois group $S_m$ is to take primes $p_1$, $p_2$ and $p_3$
with $g$ irreducible modulo $p_1$, a linear times an irreduc …
- 20.8k