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Results tagged with algebraic-number-theory
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user 4149
Algebraic number fields, Algebraic integers, Arithmetic Geometry, Elliptic Curves, Function fields, Local fields, Arithmetic groups, Automorphic forms, zeta functions, $L$-functions, Quadratic forms, Quaternion algebras, Homogenous forms, Class groups, Units, Galois theory, Group cohomology, Étale cohomology, Motives, Class field theory, Iwasawa theory, Modular curves, Shimura varieties, Jacobian varieties, Moduli spaces
3
votes
0
answers
557
views
Algebraic integer with conjugates on the unit circle
Let $\alpha$ be an algebraic integer on the unit circle in $\mathbb{C}$ such that all the conjugates of $\alpha$ lie on the unit circle. Does it follow that $\alpha$ is a root of unity?
- 14.2k
2
votes
Infiniteness of the Galois cohomology over a number field with coefficients in a finite Galo...
Let $k$ be a number field. Let $M$ be a nonzero finite $\mathrm{Gal}({\bar k}/k)$-module, i.e., a nonzero discrete finite abelian group with a continuous action of $\mathrm{Gal}({\bar k}/k)$.
Theor …
- 14.2k
2
votes
1
answer
86
views
Cyclic extensions of a number field of full local degree in a given set $S$
Let $K$ be a number field, and let $S=S_f\cup S_{\mathbb R}\cup S_{\mathbb C}$
be a finite set of places of $K$, where $S_f$ denotes the set of finite places in $S$, $S_{\mathbb R}$ denotes the set of …
- 14.2k
14
votes
3
answers
1k
views
Infiniteness of the Galois cohomology over a number field with coefficients in a finite Galo...
Let $k$ be a number field and $M$ be a nonzero finite discrete $\mathrm{Gal}(\bar k/k)$-module. Is it true that $H^1(k,M)$ is infinite?
This would complete the answer of Daniel Loughran. There is a c …
- 14.2k
4
votes
Hasse principle for rational times square
No, in general the Hasse principle for the property of being a rational number times a square does not hold. I consider the question in the form of GH from MO. I give a counter-example with a non-norm …
- 14.2k
10
votes
1
answer
537
views
A Galois extension over $\mathbb{Q}$ with Galois group $A_4$ and with cyclic decomposition g...
Does there exist a Galois extension $L/\mathbb{Q}$ with Galois group $A_4$ (the alternating group on four letters) such that all the decomposition groups are cyclic?
This question is motivated by the …
- 14.2k
3
votes
0
answers
97
views
Multiplication law in a central simple algebra of dimension 9 over a global field
Let $k$ be a global field, for example $k=\Bbb Q$.
Let $D$ denote the central simple algebra of dimension 9 over $k$ with given local invariants $i_v$.
Here $v$ runs over the set $\Omega_f(k)$ of fin …
- 14.2k
1
vote
1
answer
180
views
Decomposition groups for the Galois module $\mu_8$
$\DeclareMathOperator{\Hom}{Hom}
\DeclareMathOperator{\Aut}{Aut}
\DeclareMathOperator{\Gal}{Gal}
\newcommand{\Z}{{\Bbb Z}}
\newcommand{\Q}{{\Bbb Q}}
\newcommand{\Fbar}{{\overline F}}
\newcommand{\G}{\ …
- 14.2k
4
votes
0
answers
135
views
A normal extension of a number field of given degree that does not split over a given set of...
Let $K$ be a number field and $S$ be a finite set of non-archimedean places of $K$. Let $n>1$ be a natural number.
Question. Does there exist a normal extension $L/K$ of degree $n$ such that $L\otime …
- 14.2k
1
vote
0
answers
56
views
Normality in a tower of cyclic extensions of global fields, as in Artin-Tate
Let $L_0$ be a global field without real places, that is, a global function field or a totally imaginary number field,
and let $V_f(L_0)$ denote the set of finite (that is, non-archimedean) places of …
- 14.2k
6
votes
0
answers
167
views
Computer programs for decomposition groups?
There is quite a lot of work on computing Galois groups of splitting fields of polynomials over $\Bbb Q$. Magma is quite good at it.
In this answer to Decomposition groups for the Galois module $\mu_8 …
- 14.2k
25
votes
What is known about the cohomological dimension of algebraic number fields?
By definition, an algebraic number field is a finite extension
of the field of rational numbers $\Bbb Q$.
An algebraic number field $K$ is called totally imaginary
if it has no embeddings into $\B …
- 14.2k
7
votes
1
answer
332
views
Explicit cocycles for the first Galois cohomology of a $p$-adic torus
Let $K$ be a $p$-adic field (a finite extension of the field of $p$-adic numbers ${\mathbb Q}_p$).
Let $T$ be a $K$-torus with character group $X={\sf X}^*(T)$ and cocharacter group $Y={\sf X}_*(T)=X^ …
- 14.2k
3
votes
0
answers
94
views
Multiplication law in a division algebra of dimension 9 over a non-archimedean local field
Let $k$ be a non-archimedean local field, for example, a $p$-adic field (a finite extension of the filed ${\Bbb Q}_p$ of $p$-adic numbers).
It is well known that there is a canonical isomorphism
$${\r …
- 14.2k
1
vote
Accepted
Explicit cocycles for the first Galois cohomology of a $p$-adic torus
Answer of James S. Milne: Most probably, this homomorphism
$$\lambda\colon\, H^{-1}(\Gamma_{L/K}, Y)\overset\sim\longrightarrow H^1(\Gamma_{L/K}, Y\otimes_{\Bbb Z} L^\times)$$
is just the cup-produc …