Questions related to the algebraic structure of algebraic integers
7
votes
1answer
95 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
88 views
Set of locations where the Hilbert symbol is not equal to $1$
Let $V$ be the set of prime together with the symbol $\infty$. For a prime $v=p$, denote the $p$-adic numbers by $\mathbb{Q}_p$ and the real numbers by $\mathbb{Q}_\infty$. For $v\in V$ the Hilbert ...
7
votes
1answer
180 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
votes
1answer
191 views
How to show that $1-\zeta$ is prime in the order $\{ 1, \zeta, \ldots, \zeta^{l-2} \}$?
I am trying to prove the following:
Let $l$ be a prime and let $\zeta$ be a $l$th root of unity. Show that, in the order $\{ 1, \zeta, \ldots, \zeta^{l-2} \}$ of the field $\mathbb{Q}(\zeta)$, if ...
7
votes
1answer
102 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
votes
0answers
67 views
Numbers represented by a cubic form
EDIT, April 11, 2013: See answer at http://mathoverflow.net/questions/127160/numbers-integrally-represented-by-a-ternary-cubic-form/127295#127295
This is part 2 ( of 25 discriminants of class number ...
7
votes
0answers
109 views
CFT via Brauer groups vs via ideles
I am interested in the relationship between the following two versions of CFT:
Version 1: (Brauer Group Version)
Let $K$ be a number field. One constructs, for every finite place $v$ of $K$, a map ...
7
votes
0answers
273 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 ...
7
votes
0answers
151 views
Is the splitting field unramified over a prime $\mathfrak{p}$ if the discriminant of the polynomial is coprime to $\mathfrak{p}$?
Is the following statement true?
Let $R$ be a discrete valuation ring with quotient field $K$ and valuation $\nu$. Suppose that $f(x)\in K[x]$ is an irreducible separable polynomial with ...
7
votes
0answers
131 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 ...
6
votes
1answer
1k views
Reading the mind of Prof. John Coates (motive behind his statement)
To start with the issue, I have been thinking from many days that Birch-Swinnerton-dyer conjectures should have some association with the Galois theory, but one day I got the Article of Tate called as
...
6
votes
3answers
183 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?
6
votes
2answers
95 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
votes
5answers
906 views
Minimal polynomial of $\frac{\sqrt{2}+\sqrt[3]{5}}{\sqrt{3}}$
I am struggling to find the minimal polynomial for $\displaystyle \frac{\sqrt{2}+\sqrt[3]{5}}{\sqrt{3}}$.
Does anyone have any suggestions?
Thanks,
Katie.
6
votes
4answers
303 views
Is there a procedure to determine whether a given number is a root of unity?
Let $z$ be an algebraic number of modulus one. Is there a finite procedure that tells us whether $z$ is a root of unity?
EDIT: As TonyK and David asked, what I had in my mind is $z$ such that I have ...
6
votes
2answers
207 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 ...
6
votes
2answers
328 views
Is the algebraic closure of a $p$-adic field complete
Let $K$ be a finite extension of $\mathbf{Q}_p$, i.e., a $p$-adic field. (Is this standard terminology?)
Why is (or why isn't) an algebraic closure $\overline{K}$ complete?
Maybe this holds more ...
6
votes
2answers
216 views
Splitting of primes in an $S_3$ extension
Let $\alpha$ be a root of the polynomial $X^3-X-1$. The polynomial has discriminant $-23$ which is not a square, so the splitting field must have Galois group $S_3$. I would like to figure out the ...
6
votes
4answers
195 views
number of solutions to an equation?
Given $x$ and $y$ are multiples of $2$ satisfying
$$x^2 - y^2 = 27234702932$$
Find the number of solutions to $x$ and $y$.
6
votes
2answers
235 views
In what senses are archimedean places infinite?
According to Bjorn Poonen's notes here (§2.6), we should add the archimedean places of a number field $K$ to $\operatorname{Spec} \mathscr{O}_K$ in order to get a good analogy with smooth projective ...
6
votes
3answers
293 views
Why is a number field always of the form $\mathbb Q(\alpha)$ for $\alpha$ algebraic?
My definition of a number field is "a finite extension of $\mathbb Q$". I want to prove that if $L$ is a finite field extension of $\mathbb Q$, then $L = \mathbb Q(\alpha)$ for some $\alpha$ algebraic ...
6
votes
2answers
242 views
The Néron-Tate canonical height on elliptic curves
I have been trying to understand the Néron-Tate global canonical height of algebraic points on elliptic curves.
Let $K$ be a number field, $E$ an elliptic curve (over $\mathbb{Q}$, say), and $E(K)$ ...
6
votes
2answers
278 views
A simple question about Iwasawa Theory
There has been a lot of talk over the decades about Iwasawa Theory being a major player in number theory, and one of the most important object in said theory is the so-called Iwasawa polynomial. I ...
6
votes
2answers
336 views
Class field theory for function fields and a curious statement
Let $X_0$ be a smooth curve over a finite field $\mathbb{F}_q$, and let $X$ be the base-change to the algebraic closure. I read that, according to class field theory in function fields,
"the image ...
6
votes
3answers
128 views
Preparations for reading Algebraic Number Theory by Serge Lang
I am eager to learn algebraic number theory. It seems that Serge Lang's Algebraic Number Theory is one of the standard introductory texts (correct me if this is an inaccurate assessment). I flipped ...
6
votes
2answers
137 views
Solve: $x^2-py^2=q$
Solve $$x^2-py^2=q$$ for integers $x,y$, here $p,q$ are both given prime numbers.
It's obvious that $p,q$ should satisfy $(\frac{p}{q})=(\frac{q}{p})=1,$ here $(\frac{p}{q})$ is the Jacobi symbol.
...
6
votes
2answers
155 views
Every ideal of an algebraic number field can be principal in a suitable finite extension field
Let $K$ be an algebraic number field.
Let $I$ be a non-zero ideal of the ring of integers $\mathcal{O}_K$ in $K$.
By class field theory, there exists a finite extension(the Hilbert class field) $L$ of ...
6
votes
3answers
224 views
Every finite field is a residue field of a number field
I have read that for any finite field $\mathbb{F}_q$ there exists a number field $F$, and some prime ideal $P$ of $\mathcal{O}_F$ such that
$$\mathbb{F}_q \cong \mathcal{O}_F / P.$$
The ring of ...
6
votes
1answer
332 views
What does Tate mean when he wrote “Higher dimensional class field theory” in the new preface to the Artin-Tate book and another question?
This is probably well-known to the experts or many number theory students, but since I am just starting to learn class field theory (with some basic knowledge of algebraic numbers, e.g. the 3 basic ...
6
votes
2answers
160 views
Solving the equation by going into a non-UFD
To solve $y^2 + 2 = x^3$ you can factor $(y - \sqrt{-2})(y + \sqrt{-2}) = x^3$ and then check that they are relatively prime and by unique factorization both must be cubes then you can solve it.
What ...
6
votes
1answer
119 views
Are there infinitely many pairs of rational numbers such that…
Are there infinitely many pairs of rational numbers $(a,b)$ such that $a^3+1$ is not a square in $\mathbf{Q}$, $b^3+2$ is not a square in $\mathbf{Q}$ and $b^3+2 = x^2(a^3+1)$ for some $x$ in ...
6
votes
2answers
424 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 ...
6
votes
1answer
235 views
Lattice of Gauss and Eisenstein Integers
Z is a 1D lattice
Gaussian and Eisenstein integers are 2D lattices
But the golden integers (for example) are dense on the real line.
Are there rings of integers which have 3D, 4D, ... lattices?
...
6
votes
1answer
78 views
Ring of integers of $\mathbb{Q}(\zeta_3,\sqrt[3]{2})$.
I've seen that the ring of integers of $\mathbb{Q}(\sqrt{n})$ depends on $n\mod 4$. I am just wondering if we can (easily) write down the ring of integers of $\mathbb{Q}(\zeta_3,\sqrt[3]{2})$ (the ...
6
votes
1answer
196 views
Primes that ramify in a field
Consider the number field $L/\mathbb{Q}$. I know that the only primes $p$ that ramify over $L$ are the ones that divide $\Delta_{L}$, the discriminant of $L$. But what if I can't compute $\Delta_{L}$? ...
6
votes
3answers
249 views
Unique quad. subfield of $\mathbb{Q}(\zeta_p)$ is $\mathbb{Q}(\sqrt{p})$ if $p \equiv 1$ $(4)$, is $\mathbb{Q}(\sqrt{-p})$ if $p \equiv 3$ $(4)$
I want to prove the assertion: The unique quadratic subfield of $\mathbb{Q}(\zeta_p)$ is $\mathbb{Q}(\sqrt{p})$ when $p \equiv 1 \pmod{4}$ and is $\mathbb{Q}(\sqrt{-p})$ when $p \equiv 3 \pmod{4}$.
...
6
votes
1answer
156 views
Why is this isomorphism $M \otimes_K L \stackrel{\simeq}{\longrightarrow} M^{[L:K]}$ an isomorphism of $M$ - algebras?
Suppose that $L/K$ is a finite separable extension of fields and let $M$ denote the Galois closure of $L$. Let $\textrm{Hom}_K(L,M)$ denote the set of all $K$ - algebra homomorphisms from $L$ to $M$. ...
6
votes
1answer
162 views
Finding the units in $\mathbb{Z}[\sqrt[3]{2}]$ and other questions
I'm reading a paper in which they solve the equation: $$a^3-2b^3=\pm 1$$
in integers using algebraic number theory.
The number $a-b\alpha$, with $\alpha=\sqrt[3]{2}$, is a unit in ...
6
votes
1answer
164 views
How to compute a discriminant
Let $\alpha$ be a root of the irreducible cubic polynomial $x^{3}+px+q$, $p,q\in \mathbb{Q}$. How can I compute the discriminant $\Delta(1,\alpha,\alpha^{2})$ relative to $\mathbb{Q}(\alpha)$?
6
votes
3answers
390 views
Is there any trivial reason $2$ is irreducible in $\mathbb{Z}[\omega],\omega=e^{\frac{2\pi i}{23}}$?
This naive question came as the last problem in my homework. The author asked me to use linear relations of the discriminant like ...
6
votes
2answers
322 views
On the class group of an imaginary quadratic number field
Let $d < 0$ be a square-free integer and let $p_{1},\ldots p_{r}$ be the prime divisors of $d$. Let $K := \mathbb{Q}[\sqrt{d}]$ and consider $P_{i} := (p_{i}, \sqrt{d}) \subset \mathcal{O}_{K}$. ...
6
votes
1answer
134 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 ...
6
votes
1answer
66 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 ...
6
votes
2answers
384 views
About cyclotomic extensions of $p$-adic fields
I've been working on the problem of finding the maximal abelian extension of $\mathbb{Q}_5$ that is killed by $5$. In other words, find the abelian extension of $\mathbb{Q}_5$ with Galois group ...
6
votes
1answer
162 views
Factorization of primes and $Spec(\mathcal{O}_K)$
Let $K$ be a quadratic number field, and $\mathcal{O}_K$ the ring of integers of $K$.
The map $\pi: Spec(\mathcal{O}_K) \rightarrow Spec(\mathbb{Z})$ that sends a prime ideal $\mathbb{p}$ to ...
6
votes
1answer
162 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 ...
6
votes
1answer
130 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 ...
6
votes
1answer
281 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
...
6
votes
1answer
118 views
$\mathbb Z_p$-rank $\leq r_1+r_2-1$ in Leopoldt's conjecture
I am trying to understand Leopoldt's conjecture as formulated in section 5.5 of Washington's "Introduction to Cyclotomic Fields".
For an algebraic number field $K$ let $E$ denote the global units, ...
6
votes
1answer
398 views
Class number computation (cyclotomic field)
How does one prove that the class number of $\mathbb{Q}(\zeta_{23})$ is divisible by $3$? And afterwards how do you show that it is precisely $3$. Any help?
Thanks in advance!
//Ok, so I proved the ...
