Questions related to the algebraic structure of algebraic integers

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7
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1answer
120 views

Infinite $p$-extension contains $\mathbb{Z}_p$-extension

Does the Galois group of every infinite $p$-extension $K$ of a number field $k$ contain a (closed) subgroup such that the quotient group is isomorphic to $\mathbb{Z}_p$? My feeling is "yes", but I'm ...
7
votes
1answer
216 views

Residue at $s=1$ for $\zeta$-functions

Is there any sort of a bound for the magnitude of this residue? I've been looking at some algebraic number theory problems from Princeton's general exam. One of them is the following: Why does a ...
7
votes
1answer
361 views

why does a certain formula in Lang's book on modular forms hold?

Background: Let $k$ be an even integer. The Eisenstein series are defined by $$E_{k} = 1 - \frac{2k}{B_{k}}\sum_{n=1}^{\infty} \sigma_{k-1}(n)q^{n}$$ where $$\sigma_{k-1}(n)= \sum\limits_{d \mid ...
7
votes
2answers
542 views

A question on FLT and Taniyama Shimura

Sometime back i watched the documentary of Andrew Wiles proving the Fermat's Last theorem. A truly inspiring video and i still watch it whenever i am in a depressed mood. There are certain ...
7
votes
1answer
165 views

How does (21) factor into prime ideals in the ring $\mathbb{Z}[\sqrt{-5}]$?

The text of the exercise is the following: Show that $\mathbb{Z}[\sqrt{-5}]$ is a Dedekind domain, and that the identities $21 = (4+\sqrt{−5}) \cdot (4 − \sqrt{−5})$ and $21 = 3 · 7$ represent two ...
7
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1answer
377 views

How to arrive at Ramanujan nested radical identity

I've come across a very curious Ramanujan identity $$\sqrt[6]{7 \sqrt[3]{20} - 19} = \sqrt[3]{\frac{5}{3}} - \sqrt[3]{\frac{2}{3}}$$ You could probably prove this by taking the 6th power of both ...
7
votes
1answer
318 views

What is the best book learn Galois Theory if I am planning to do number theory in future?

What is the best book learn Galois Theory if I am planning to do number theory in future? In a year i'll be joining for my Phd and my area of interest is number theory. So I want to know if there is ...
7
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2answers
771 views

Ideal class group of $\mathbb{Q}(\sqrt{-103})$

I want to calculate ideal class group of $\mathbb{Q}(\sqrt{-103})$. By Minkowsky bound every class has an ideal $I$ such that $N(I) \leq 6$. It is enough to consider prime ideals with the same ...
7
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1answer
988 views

What is the discriminant of a quadratic extension over a number field?

Let $K$ be a number field and $d \in \mathcal{O}_K \setminus \mathcal{O}_K^2$. What is the discriminant of the extension $K[\sqrt{d}]/K$ ? Do we know its ring of integers and which primes are split or ...
7
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2answers
110 views

Primes inert in quadratic field of class number one

Let $K = \mathbb{Q}(\sqrt{-d})$ be an imaginary quadratic field of class number one (i.e. every ideal in $\mathcal{O}_K$ is principal, i.e. $\mathcal{O}_K$ is a principal ideal domain). Let $d_K$ be ...
7
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1answer
126 views

can anyone give a proof by definition: $11$ is prime in $ \mathbb{Z}[\sqrt{-5}] $

what i did is: assume $\alpha \notin (11),\beta\notin (11), \alpha\beta \in (11)\Rightarrow\exists \gamma, s.t.$ $ \alpha\beta = 11 \gamma$, $\Rightarrow N(\alpha)N(\beta) = 11^2N(\gamma) $ then ...
7
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1answer
79 views

Generalizing the Big Omega function to Integral Domains

The $\Omega(n)$ function counts the total number of prime factors of $n$ counting multiplicity. Obviously, this definition extends to any Unique Factorization Domain. I have two follow up questions: ...
7
votes
1answer
306 views

Computing Brauer characters of a finite group

I am studying character theory from the book "Character Theory of Finite Groups" by Martin Isaac. (I am not too familiar with valuations and algebraic number theory.) In the last chapter on modular ...
7
votes
1answer
247 views

All number fields with absolute value of discriminant $\le 20$

I need to find all number fields with absolute value of discriminant $\le 20$. Using Minkovsky's theorem I understood that it should be quadric or cubic extension. The case of quadric is very ...
7
votes
1answer
303 views

Computing the ring of integers of a number field

This question arose from the need to see the splitting behaviour of primes over an extension of number fields. One criterion is the Kummer's theorem, which gives the decomposition of the base prime in ...
7
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1answer
294 views

Ring class field of $\mathbb{Q}(\sqrt{-19})$

I am looking an explicit form for the ring class field of the order $\mathbb{Z}[\sqrt{-19}]$ in the quadratic field $\mathbb{Q}(\sqrt{-19})$. Does anyone know if there is some and how it is?
7
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1answer
154 views

p-adic modular form example

In Serre's paper on $p$-adic modular forms, he gives the example (in the case of $p = 2,3,5$) of $\frac{1}{Q}$ and $\frac{1}{j}$ as $p$-adic modular forms, where $Q = E_4 = 1 + 540\sum ...
7
votes
1answer
108 views

Proof in Kummer Theory - why is this subgroup finite?

I'm doing a course on elliptic curves. We're working with a field $K$ with $\mu_n \subset K$ ($K$ contains all $n$th roots of unity) and $\mbox{char}(K)\not|\;n$. I'm trying to understand the proof ...
7
votes
1answer
139 views

Ramification indices and residue degrees of a finite Galois extension

Let $A$ be a dvr with fraction field $K$ of characteristic zero. Let $L/K$ be a finite Galois extension and let $B$ be the integral closure of $A$ in $L$. For a prime $b$ of $B$, let $e_b$ be its ...
7
votes
1answer
171 views

Determining when a certain binomial sum vanishes

Consider the following sum of signed binomial coefficients: $$S_{n,a,p} = \sum_{i \equiv a \mod p} \binom{n}{i}(-1)^i$$ ($n$ is a positive integer, $p$ is an odd prime, $a$ is between $0$ and ...
7
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1answer
186 views

How can I determine in practice whether two elliptic curves over $\mathbb{Q}$ have isomorphic $p$-torsion?

Let $E_1$ and $E_2$ be elliptic curves over $\mathbb{Q}$ with good, ordinary reduction at an odd prime $p$. I'm wondering how to determine whether $E_1[p]$ and $E_2[p]$ are isomorphic ...
7
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1answer
358 views

Special privilege enjoyed by Elliptic Curves with Complex Multiplication

I think after reading the title one may understand the intention of me, this question is concerned about the Elliptic curves having a Complex Multiplication. I have been reading many theorems, ( ...
7
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1answer
928 views

Ramification index and inertia degree

Let $L,K$ be number fields and $L|K$ a galois extension. Let $(0)\neq Q$ a prime ideal in $\mathcal O_L$ (=ring of integers in $L$) and $P=Q \cap \mathcal O_K$. $Z_Q $ denotes the decomposition ...
7
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1answer
112 views

Implications between $(\frak{a} \cap \frak{b})$ = $\frak{a} \frak{b}$ and $(\frak{a})$ + $(\frak{b})$= $(1)$

In a general commutative ring, $(\frak{a} \cap \frak{b})$ = $\frak{a} \frak{b}$ does not imply ($\frak{a}$) + ($\frak{b}$) = ($1$); whereas ($\frak{a}$) + ($\frak{b}$) = ($1$) does imply $(\frak{a} ...
7
votes
1answer
258 views

Doesn't Abhyankar's lemma contradict faithful flat descent (no, but I'm confused)

Let $O_K$ be a dvr with fraction field $K$. Let $L/K$ be a tamely ramified finite Galois extension. Then, Abhyankar's Lemma implies that there exists a finite Galois extension $K^\prime/K$ such that ...
7
votes
1answer
128 views

“Real part” of a number field

Let ${\mathbb K} \subseteq {\mathbb C}$ be a finite extension of $\mathbb Q$, and let $n=[{\mathbb K} : {\mathbb Q}]$. Let $X_{\mathbb K}$ denote the set of all “components” (i.e., real and imaginary ...
7
votes
1answer
186 views

The number of solutions of the equation $x^n+y^n=1$ over $\mathbb{Z}_p$ with $\mathbb{Z}_p=\{\alpha\in \mathbb{Q}_p:|\alpha|_p\le 1\}$.

The number of solutions of the equation $x^n+y^n=1$ over $\mathbb{Z}_p$ with $\mathbb{Z}_p=\{\alpha\in \mathbb{Q}_p:|\alpha|_p\le 1\}=\{\sum_{i=0}^\infty a_ip^i:0\le a_i \le p-1\}$. I was trying ...
7
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1answer
470 views

How to show this ideal is not principal

I have been brushing up on cubic number fields. Specifically, let $s$ be a root of the polynomial $x^3 + x^2 + 3x + 17$, and consider $K = \mathbb{Q}(s)$; we have $\mathcal{O}_K = \mathbb{Z}[s]$, and ...
7
votes
1answer
61 views

An extension of an algebraic number field which makes an integral ideal $I$, a principal ideal

I want to show that, given an ideal $I \subseteq \mathcal O_K$ (where $K/\mathbb Q$ is an algebraic number field), there is a finite extension $K'/K$ such that, $I\mathcal O_{K'}$ becomes a principal ...
7
votes
1answer
128 views

Tate's Thesis: Meaning of Local Functional Equation

I am studying the development of Tate's Thesis in Lang's Algebraic Number Theory and have a conceptual question. The setting: Let $k=\mathbb{Q}_p$. Let $\mu$ be the unique Haar measure giving ...
7
votes
2answers
536 views

Bijection between an ideal class group and a set of classes of binary quadratic forms.

Let $F = ax^2 + bxy + cy^2$ be a binary quadratic form over $\mathbb{Z}$. We say $D = b^2 - 4ac$ is the discriminant of $F$. If $D$ is not a square integer and gcd($a, b, c) = 1$, we say $ax^2 + bxy + ...
7
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1answer
237 views

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 ...
7
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1answer
112 views

Computing a uniformizer in a totally ramified extension of $\mathbb{Q}_p$.

Do you know how to compute a uniformizer of $\mathbb{Q}_p(\zeta_{p^n},p^\frac{1}{p})$? Where $\zeta_{p^n}$ is a primitive $p^n$-th root of 1 and $p$ is an odd prime.
7
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1answer
70 views

probability that two randomly selected integers of an imaginary quadratic field of class number 1 are coprime

Given an imaginary quadratic field $\mathbb{Q}(\sqrt{-D})$, where $D$ is a Heegner number (1, 2, 3, 7, 11, 19, 43, 67, 163), what is the probability that two randomly selected elements of that fields' ...
7
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1answer
133 views

Characterizing a sequence of primes

This is an attempt to finish up Characterizing the primes which don't divide any Pell-Lucas number(s) For primes $p \equiv 3 \pmod 4,$ there is always some solution to $x^2 - 2 y^2 = \pm 1$ with ...
7
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1answer
236 views

Degrees of the real and imaginary parts of an algebraic number

I am working on a theory of generalized geometric constructions, which involves generating new numbers as real roots of polynomials whose coefficients are existing numbers satisfying certain ...
7
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1answer
230 views

Show the two fields are not isomorphic

Let $p,q,r$ be prime integers with $q\neq r$. Let $\sqrt[p]q$ denote any root of $x^p-q$ and $\sqrt[p]r$ denote any root of $x^p-r$. Please show that $\mathbb{Q}(\sqrt[p]q)\neq\mathbb{Q}(\sqrt[p]r)$. ...
7
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1answer
142 views

Proof on p. $16 \;$ of Lang's Algebraic Number Theory

I'm trying to understand the proof of Corollary $1$ on p. $16$. It says Corollary $1$. Let $A$ be a ring integrally closed in its quotient field $K$. Let $L$ be a finite Galois extension of $K$, ...
7
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1answer
309 views

How does the Artin symbol generalize Legendre and Hilbert symbols?

I am currently going through a course in Class Field theory and have read that the Artin symbol generalizes the Legendre and Hilbert symbols. I was wondering what field extensions to consider to see ...
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1answer
99 views

Volume of first cohomology of arithmetic complex

Let $K$ be a number field and consider the Arithmentic complex $\Gamma_{Ar}(1)^\bullet$ be defined by $$\begin{array} A\Bbb R^{r_1+r_2} & \stackrel{\Sigma}{\longrightarrow} & \Bbb R \\ ...
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0answers
177 views

Adelic/Idelic method for number fields

I'm searching for a place where is developed all the machinery of adeles and ideles of a number field; the functional equation for the zeta functions is gained by idelic integration, and it's seen the ...
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0answers
81 views

Ex. $2.30$ in Silverman Adv. Topics

I would like to refer you to Exercise $2.30(c)$ in Silverman's Advanced Topics in Elliptic Curves. Question: Let $E/L$ be an EC with CM by $K$. Assume that $K\nsubseteq L$, and let $L'=LK$ and let ...
7
votes
2answers
126 views

Hilbert class field application

If $K$ is an imaginary quadratic field and $M$ is an unramified Abelian extension of $K$, the prove that $M$ is Galois over $\mathbb{Q}$ Let see...If $L$ is the Hilbert class field of $K$, then $L$ ...
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0answers
486 views

A conjecture about Lucas series

Let $L_n$ be the Lucas series: $L_0=2,L_1=1,L_n=L_{n-1}+L_{n-2}(n>1).$ $p$ is a prime number and $p\equiv3,7\pmod {20}$, hence $\exists x,y\in \mathbb Z:2p=x^2+5y^2.$ Is it true that ...
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0answers
485 views

Compute the Unit and Class Number of a pure cubic field $\mathbb{Q}(\sqrt[3]{6})$

Find a unit in $\mathbb{Q}(\sqrt[3]{6})$ and show that this field has class number $h=1$. I am done with the first part which is relatively simple: Suppose that $\varepsilon$ is a unit in ...
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0answers
374 views

notation for ramification index and inertial degree

For a prime $Q$ lying over a prime $P$, I have seen the ramification index of $Q$ over $P$ denoted by $e(Q|P)$ and the inertial degree of $Q$ over $P$ by $f(Q|P)$. What is the origin of the ...
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0answers
150 views

closure of units of number fields in the finite idele topology

Let $K$ be a number field. Denote by $\mathcal O _K^\times$ its rings of units and by $\mathcal O _{K,+} ^\times$ its ring of totally positive units. Further let us denote by $\mathbb A ...
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votes
2answers
145 views

Usage of algebraic geometry in understanding the total Galois group of the rational

A professor of mine remarked in class that: "The tools via which we understand the total Galois group of the rationals comes from algebraic geometry". Could anyone shed some light on this remark, or ...
6
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3answers
241 views

What's the difference between $\mathbb{Q}[\sqrt{-d}]$ and $\mathbb{Q}(\sqrt{-d})$?

Sorry to ask this, I know it's not really a maths question but a definition question, but Googling didn't help. When asked to show that elements in each are irreducible, is it the same?
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2answers
249 views

Galois extension of $\mathbb{Q}_2$ with Galois group $(\mathbb{Z}/2\mathbb{Z})^3$

I am trying to solve this exercise: Prove that $\mathbb{Q}_2$ has a unique Galois extension F with Galois group $(\mathbb{Z}/2\mathbb{Z})^3$. Compute it's ramification groups. Here is what I have ...