Questions tagged [class-field-theory]

Class field theory is a major branch of algebraic number theory that studies abelian extensions of global and local fields.

685 questions
Filter by
Sorted by
Tagged with
61 views

Examples of Lubin-Tate formal groups

I'm currently learning about Lubin-Tate theory: Given a p-adic field $K$ and a uniformizer $\pi \in \mathcal{O}_K$, they consider formal $\mathcal{O}_K$-modules $F$ such that the endomorphism $[\pi]$ ...
• 83
36 views

Norm surjective for unramifeid extension of local fields $L/K$

I have a question about the argument used here in proof of Proposition 4.2.5.: For any finite unramified extension $L/K$ of local fields, the map $\text{Norm}_{L/K}: O_L^* \to O_K^*$ is surjective. ...
• 7,489
49 views

Are Hecke L-functions associated to Artin L-functions primitive?

I'm reading a famous paper by Lagarias & Odlyzko on effective versions of the Chebotarev density theorem. There is one thing about Artin L-functions that is a little perplexing to me. Let $L/K$ be ...
• 769
32 views

Is there any relationship between the study of class number of a field and the study of class field theory through Lubin-Tate formal group?

I am curious to know if one can relate to the study of Lubin-Tate formal group and related local class field theory with the study of class number of a field (global field in general). As far as I ...
• 10.8k
59 views

• 179
27 views

class number of quadratic field with discriminant $17$ [duplicate]

I'm trying to prove that class number of the quadratic field with discriminant $17$ is $1$. Let $K =Q[\sqrt{17}]$ be the quadratic field and $\mathfrak{o}_{K}=\mathbb{Z}[\frac{1+\sqrt{17}}{2}]$ be the ...
105 views

Show by Local Class Field Theory $\mathbf Q_p$ has unique Galois ext. iso. to $(Z/2Z)^2$ if $p > 2$, and unique Galois ext. iso. to $(Z/2Z)^3$ o/w.

Show that $\mathbf{Q}_p$ has a unique Galois extension isomorphic to $(Z/2Z)^2$ if $p > 2$, and that $\mathbf{Q}_2$ has a unique Galois extension isomorphic to $(Z/2Z)^3$ I have already completed ...
• 391
1 vote
87 views

Prove that for a imaginary quadratic field $K/\mathbb Q$, $Cl(K)/2Cl(K)= (\mathbb Z/2)^{r-1}$ where $r$ is the number of prime that ramify in $K$.

I want to prove that for a imaginary quadratic field $K/\mathbb Q$, $Cl(K)/2Cl(K)\simeq (\mathbb Z/2)^{r-1}$ where $r$ is the number of prime that ramify in $K$. This question has appeared on this ...
• 3,007
59 views

Find an abelian extension of 2-adic rationals such that $Gal(K/\mathbb Q_2)= \mathbb Z/2 \times (\mathbb Z /2^r)^2$.

Let $r>0$. I want to prove that there exists a extension $K / \mathbb Q_2$ such that $Gal(K/\mathbb Q_2)= \mathbb Z/2 \times (\mathbb Z /2^r)^2$. This is an exercise in Kedlaya's notes on class ...
• 3,007
72 views

Lang exercise 50 on Witt vectors

I'm reading the construction of the Witt ring from Lang's algebra. This is a series of exercises in chap. VI. In exercise 50 he says: If $x\in W_n(k)$ show that there exists $\xi\in W_n(\bar{k})$ ...
42 views

• 2,058
86 views

"Classical" approach to local class field theory (Brauer group and Hasse invariant)

I'm trying to learn local class field theory from the corresponding chapter of Lorenz's "Algebra 2", which, according to its mission statement, takes a more classical approach [Local class ...
• 1,434
1 vote
37 views

Normal extension of given degree of a number field

Let $n>1$ be a natural number. Question. Let $K$ be a number field, and let $S\subset V_f(K)$ be a finite set of finite (non-archimedean) places of $K$. Does there exist a normal extension $L/K$ ...
• 1,226
1 vote
154 views

Maximal p-extension and pro-p extension

I’m studying Iwasawa theory and I meet some questions Thanks a lot for your help. Q_1: About terminology $p$-extension. I find many reference use maximal $p$-extension or maximal abelian p-extension ...
• 13
40 views

Formal groups over discrete valuation rings

Let $k/\mathbb{Q}_p$ be a finite extension. Let $k^{ur}$ be its maximal unratified extension and consider $K$ its completion. Let $\varphi$ be the Frobenius morphism in Gal$(k^{ur}/k)$ and whit the ...
• 739
33 views

• 417