All questions with tag [math: algebraic-number-theory]


Why is the restricted direct product topology on the idele group stronger than the topology induced by the adele group?

I am confused by the saying that the restricted direct product topology on the idele group is stronger than the topology induced by the adele group. And perhaps this is elaborated by the following example Let $p_n$ be the $n$-th positive prime in $\mathbb{Z}$, and let $\alpha^{n}=(\alpha^{(n)}_v)\in\mathbb{A}_\mathbb{Q}$ with $\alpha^{(n)}_v...
2022-07-25 17:44:17

DVR-valued points of schemes

Let $X$ be a scheme of finite type over a discrete valuation ring $R$ with fraction field $K$, such that the generic fibre $X_K$ is smooth over $K$. Let $Y$ be a closed subscheme of $X$ which contains no irreducible component of $X$. Is it true - maybe under some additional assumptions on $X$ and/or $R$ - that $X(R) \setminus Y(R)$ is dense in $...
2022-07-25 17:42:46

Solving a Diophantine Equation using factorisation of ideals

I am stuck on the following question which is given as follows: Prove that the only integer solutions to the equation \begin{equation} x^2 + 13 = y^3 \end{equation} are $(70,17)$ and $(-70, 17)$. (Hint: first show that $x$ is even and $y$ is odd) I have seen a solution to this and first part to it is as follows. Let $x$ and $y$ satisfy the given...
2022-07-25 17:13:59

The degree of finite field extensions.

I am revising for my algebraic number theory exam and was hoping I could get on some help on the following two questions: a) Let $\alpha$ be algebraic over a field $K$ of odd degree. Show that $K(\alpha) = K(\alpha^2)$ b) Let $L = K(\alpha, \beta)$, with deg$_K(\alpha) = m, \, $ deg$_K(\beta) = n, \,$ and gcd$(m,n) = 1$. Show that $[L : K] = mn...
2022-07-25 17:02:35

A proof of a theorem on the different in algebraic number fields

Theorem Let $K ⊂ L ⊂ E$ be a tower of algebraic number fields. Suppose that $E/K$ is a Galois extension. Let $B$ and $C$ be the rings of algebraic integers in $L$ and $E$ respectively. Let $G$ be the Galois group of $E/K$. Let $H$ be the Galois group of $E/L$. Let $G/H$ be the set of left cosets of $G$ by $H$. Let $S$ be the set of representativ...
2022-07-25 16:58:01

what's the conductor of a ray class field?

Let $K$ be a number field. The theorems of class field theory tell us that given any modulus $\mathfrak{m}$ for $K$, there is a unique Abelian extension $K_{\mathfrak{m}}$ such that the kernel of the Artin map of $K_{\mathfrak{m}}/K$ with respect to $\mathfrak{m}$ is precisely the subgroup of principal fractional ideals congruent to $1 \pmod{\ma...
2022-07-25 16:56:08

Question about a proof on p70 in Cassels' Local Fields

I'm trying to read the proof of COROLLARY. The only solutions of $x^2+7=2^m$ ($x,m \in \mathbb{Z}$) (6.15) have $m=3,4,5,7,15$. I don't see why there could be a + in $y\pm \alpha$ (6.17) or in $y \pm \beta$ (6.18): PROOF. Clearly $x$ is odd, say $x=2y-1$ ($y \in \mathbb{Z}$) and then $y^2-y+2=2^{m-2}$ (6.16) The ring $\mathbb{Z}[\alpha]$, whe...
2022-07-25 16:38:44

Approximation Lemma in Serre's Local Fields

Let $A$ be a Dedekind domain, and let $K$ be its field of fractions. In Serre's Local Fields, the following Lemma is stated. Approximation Lemma Let $k$ be a positive integer. For every $i$, $1\leq i \leq k$, let $\mathfrak p_i$ be prime ideals of $A$, $x_i$ elements of $K$, and $n_i$ integers. Then there exists an $x\in L$ such that $v_{\mathf...
2022-07-25 12:52:55

What is a maximal abelian extension of a number field and what does its Galois group look like?

How does one know that a number field $K$ has a maximal abelian extension (unique up to isomorphism) $K^{\text{ab}}$? I've read proofs involving Zorn's lemma that it has an algebraic closure (And that algebraic closures are unique up to isomorphism.) $\bar{K}$ All these proofs involved ideals of the polynomial ring in variables $x_f$, $f$ an i...
2022-07-25 12:34:03

How do we know if there are any better bounds than the Minkowski bound?

This question may be an exact replicate of some earlier question elsewhere. I want to know if there are better bounds than the Minkowski bound for the norm of the smallest ideal in an ideal class of a number field, which is effective when $n$ is small but seems not very sharp when $n$ is large. I noticed the crucial part of Minkowski's bound co...
2022-07-25 07:39:50

Group of finite ideles

Let $K$ be a number area. Allow $\mathbb{I}_f$ represent the team of limited ideles, and also allow $\phi: K^{\times} \rightarrow \mathbb{I}_f$ be the angled embedding. On web page 167 of his notes on Class Field Theory, Milne mentions the adhering to outcome without proof: "The generated geography on $K^{\times}$ has the adhering to summ...
2022-07-24 06:22:05

Endomorphism ring of the formal additive group law

Throughout I am looking at one parameter group laws. Let $R$ be a commutative ring with identity. Let $\mathbb{G}_a(X,Y) \in R[[X,Y]]$ be the formal additive group law, i.e., $\mathbb{G}_a(X,Y)= X+Y$. I proved that $\operatorname{End}(\mathbb{G}_a) \cong R$, when $R$ has characteristic $0$. I am trying to figure out what $\operatorname{End}(\ma...
2022-07-24 02:51:40

Global class field theory without p-adic method

I recognize the p - adic method is necessary in algebraic number theory. Nonetheless, in the old days, the international class area concept was created making use of just perfects and also timeless analysis. I'm interested to find out about it. An additional factor is that I assume the excellent logical strategy is extra constructible than the p...
2022-07-22 15:19:33

What are the branches of the $p$-adic zeta function?

I'm reading the book $p$-adic Numbers, $p$-adic Analysis, and Zeta-Functions by Neal Koblitz. In it, Koblitz wants to iterpolate the Riemann Zeta function for the values $\zeta_p(1-k)$ with $k \in \mathbb{N}$, where $\zeta_p(s) = (1-p^s)\zeta(s)$. Doing this, he defines the next function: $$ \zeta_{p,s_0}(s) = \frac{1}{\alpha^{-(s_0+(p-1)s)}-1}...
2022-07-22 15:17:16

$\mathbb{Q}(\sqrt{d})$ with specific integral basis

I would like some help with the following question. Ireland and Rosen (ch.13#10) For which $d$ does $\mathbb{Q}(\sqrt{d})$ have an integral basis of the form $\alpha, \alpha '$ where $\alpha '$ is the conjugate of $\alpha$? As I understand this, $a,b\sqrt{d}$ is a basis for $\mathbb{Q}(\sqrt{d})$, so we can set $a+b\sqrt{d}=a_{1}\alpha +b_{2}\...
2022-07-22 15:13:14

Extending Galois automorphism to group automorphism

Let $F \subset K$ be a field extension of degree $n$, then $F^n \cong K$ as $F$-vectorspaces. Now $K^\times$ acts on $F^n$, by multiplication on $K$, and so $K^\times$ embeds into $GL_n(F)$, and every Galois element gives an automorphism of $K^\times$. Question: Under which conditions can it be extended to an automorphism of $GL_n(F)$? How? Cy...
2022-07-22 15:09:50

Ideal theoretic proof of the second inequality of global class field theory

Classically the 2nd (or the first in the old terms) inequality of international class area concept ($≦ [L : K]$, see, as an example, the Milne is training course note) was confirmed making use of Zeta features and also L features. Modern evidence make use of ideles and also team cohomology. Exists an evidence of the 2nd inequality making use of ...
2022-07-22 15:09:00

The polynomial $x^p-x-1/p$ over $\mathbb Q_p$

I know that the polynomial $f(x)=x^p-x-\frac1p\in\mathbb Q_p[x]$ is irreducible. So, let $\alpha$ be a root of $f(x)$, and $K=\mathbb Q_p(\alpha)$. Let $O_K$ be the valuation ring of $K$ with respect to the unique extension of the p-adic valuation to $K$. Let $\mathfrak p_K$ be the unique maximal ideal of $O_K$. We know that $O_K$ is also the in...
2022-07-22 12:43:42

Newton polygons

This question is primarily to clear up some confusion I have about Newton polygons. Consider the polynomial $x^4 + 5x^2 +25 \in \mathbb{Q}_{5}[x]$. I have to decide if this polynomial is irreducible over $\mathbb{Q}_{5}$. So, I compute its Newton polygon. On doing this I find that the vertices of the polygon are $(0,2)$, $(2,1)$ and $(4,0)$. Th...
2022-07-22 12:42:53

Class number and Narrow Class Number

It is well recorded that there are 9 fictional square number areas with class number 1 and also therefore they have slim class number 1 too. Nonetheless, exist fictional square number areas with class number > 1 and also slim class number 1? If so, exist great deals of them?
2022-07-22 12:39:00