Questions tagged [algebraic-number-theory]
Questions related to the algebraic structure of algebraic integers
7,885
questions
0
votes
0
answers
11
views
Understanding discriminant of composite field via representation theory
Let $L, K$ be two number fields. If $L$ and $K$ are linearly disjoint over $\mathbb{Q}$, then we know (by [Neukirch, ch.I (2.11), Proposition])
$$
d_{LK}=d_L^{[LK:L]}d_K^{[LK/K]}
$$
where $d_K$ stands ...
- 1,346
0
votes
1
answer
12
views
Find the number of order triples (x, y, z) of integers?
Question:
Find the number of order triples x, y, z of integers such $that
$x^4+y^4+z^4-2x^2y^2-2y^2z^2-2z^2x^2=24$
My try:
$(x^2-y^2-z^2)^2=x^4+y^4+z^4-2x^2y^2+2y^2z^2-2x^2z^2$
So, $x^4+y^4+z^4-2x^2y^...
- 37
2
votes
1
answer
53
views
What do we know about integer solution(s) to $x^p+ay^p=1$, where $p$ is a prime number and $a$ is a fixed integer?
We know how to write the solution to the equation $x^2+ay^2=1$.
Due to a Theorem by Nagell the solution to $x^3+ay^3=1$ has at most two solutions.
What do we know about $x^p+a y^p=1$. Does this have ...
- 420
-1
votes
1
answer
63
views
when $x^2+2$ splits $\operatorname{mod}p$
Why $x^2+2$ splits iff $p \equiv 1$ or $3 \bmod 8$.
It is easy to check $p \equiv 1 \bmod 8 \quad x^2+2$ splits. In fact, if $\mathbb{F}_p$ contains a 8 th root of 1. Then $\left(\xi-\xi^{-1}\right)^2+...
- 37
3
votes
1
answer
76
views
kernel of $K[X,Y] \rightarrow K[\theta_1, \theta_2]$
Given a field $K$, for example the field of rational numbers ${\bf Q}$, and two algebraic numbers $\theta_1$, $\theta_2$ over $K$, what is the kernel of the morphism $$K[X,Y] \rightarrow K[\theta_1, \...
- 6,000
0
votes
1
answer
43
views
an isomorphism of the residue class field
I was reading first chapter of Cassel's and Frolich ANT and then I came across the following proposition : $k \cong p^n/p^{n+1}$ of k-modules.
In the first line of the proof it says if $p = R\pi$ then ...
1
vote
1
answer
44
views
Proving $\left[a_1,\frac{b_1+\sqrt\Delta}2\right]\cdot\left[a_2,\frac{b_2+\sqrt\Delta}2\right]=\left[a_1a_2,\frac{b+\sqrt\Delta}2\right]$
Let $F=\mathbb Q(\sqrt{\Delta})$ be a quadratic field and $\mathcal O$ a quadratic order of $F$ with discriminant $\Delta$. Let
$$\mathfrak{a}_1=\left[a_1,\frac{b_1+\sqrt{\Delta}}{2}\right],\quad \...
- 925
1
vote
2
answers
58
views
For what number rings does $N^{1/d}$ form an absolute value?
If $K/\mathbb{Q}$ is a number field of degree $d$ with ring of integers $\mathcal{O}_K$ when is $f:=|N|^{1/d}$ an absolute value on $\mathcal{O}_K$? Here $N$ is the algebraic norm that sends $\xi \in \...
- 778
4
votes
1
answer
110
views
When and how does a rational prime (not ideal!) reduce in a given number field?
I apologize if my abstract algebra is so shaky that I might have glossed over an answer to this problem.
Yesterday I suddenly got the motivation to do some number theory again - as a challenge, I ...
- 1,458
10
votes
1
answer
150
views
Does minimal polynomial being separable mod $\mathfrak p$ imply that $p$ does not divide $\left|\mathcal O_L / \mathcal O_K[\alpha]\right|$?
Let $L/K$ be a Galois extension of number fields, where $L=K(\alpha)$ for some algebraic integer $\alpha\in\mathcal O_L$. Let $f(x)$ be the minimal polynomial of $\alpha$ over $K$. Let $\mathfrak p$ ...
- 7,111
11
votes
6
answers
548
views
Irreducibility of $x^8+6x^4+1$
Is there a simple way to show that the polynomial $P(x) = x^8+6x^4+1$ is irreducible over $\mathbb Q$ by hand? The CAS software Sage tells me that it indeed is. It appears as polynomial 8.0.4194304.1 ...
- 6,000
1
vote
1
answer
58
views
Question about Neukirch ANT Proposition $4.2$
A subgroup $\Gamma\subset V$ is a lattice if and only if it is discrete.
The proof is on page 25 there are few non trivial lines for me
As the $\mu_i= \gamma_i- \gamma_{0i}\in \Gamma$
Why $\mu_i $ ...
- 648
2
votes
1
answer
83
views
Primes decomposition in intermediate field
let $F/K$ be a Galois extension, and $L=K(\alpha) $ be an intermediate field, $\alpha$ is an algebraic number of $K$. Let $G=\text{Gal}(F/K)$. and let $H=\text{Gal}(F/L)$, and let
$X:=\{ \text{the ...
- 662
5
votes
1
answer
102
views
Metrisability of Adele ring
Let $\mathbb{A}_{\mathbb{Q}}$ denote the topological ring of adeles over the field of rational numbers. Is its topology metrisable? If so, is it complete, separable under that metric?
I believe that ...
- 1,249
7
votes
1
answer
120
views
Is adding golden ratio to Eisenstein integers still a PID?
Let $ \omega=e^{2 \pi/3} $ be a primitive third root of unity. And let $ \phi= \frac{1+\sqrt{5}}{2} $ be the golden ratio.
The ring of integers $ \mathbb{Z}[\omega] $, called the Eisenstein integers, ...
- 8,796
0
votes
1
answer
30
views
$I \sigma (I)$ is a principal ideal in a quadratic number ring
I started learning about number theory and I can't understand this exercise that was left in lecturer's notes.
Exercise: Let $K$ be a quadratic number field and $\sigma: K \to K$ a nontrivial $\mathbb{...
- 618
0
votes
0
answers
36
views
Can you deduce the size of a residue field of a number field with a $P$-adic norm based on its degree and $p$ that $P$ lies over?
Let $K$ be an algebraic number field with a $P$-adic norm and $\mathcal{O}_K$ it's ring of integers. We know that by definition $K$ is an extension of $\mathbb{Q}$ of degree $n$. We also know that $P$ ...
- 81
0
votes
1
answer
49
views
Doubt on Neukirch ANT $(3.1)$ theorem .
On the page 17 of neukirch ANT i have a doubt in the proof
The ring $\mathcal{O}_K$ is noetherian, integrally closed, and every prime ideal $\mathcal{p}\neq0$ is a maximal ideal
In the proof while ...
- 648
4
votes
0
answers
210
views
+100
Doubts in Silverman's AEC Chapter 3 Prop 1.5
On Page $48$ of Silverman's Arithmetic of Elliptic Curves, he proves the theorem that the invariant differential associated to a Weierstrass equation for an elliptic curve is holomorphic and ...
1
vote
1
answer
65
views
Error in proof of Childress Class Field Theory, Theorem 7.2
Let $E = \mathbb{Q}(\zeta_p)$, and $E^+ = \mathbb{Q}(\zeta_p + \zeta_p^{-1})$. Let $C_K$ denote the class group of a number field $K$. Theorem 7.2 of Childress states that the map $C_{E^+} \to C_E$ ...
- 2,170
2
votes
1
answer
48
views
Is $K^\times/F^\times N_{L/K}(L^\times)$ infinite when $L/F$ is a biquadratic extension of a number field $F$?
Let $F$ be a number field, let $L=F(\sqrt{\alpha_1},\sqrt{\alpha_2})$ be a biquadratic extension, and let $K=F(\sqrt{\alpha_1\alpha_2})$.
Question. Is is true that $K^\times/F^\times N_{L/K}(L^\times)$...
- 15.4k
0
votes
0
answers
25
views
Existence of prime ideals with given splitting in biquadratic extensions
Let $F$ be a number field, let $L=F(\sqrt{\alpha_1},\sqrt{\alpha_2})$ be a biquadratic extension of $F$ (so $\alpha_1,\alpha_2$ and $\alpha_1\alpha_2$ are not squares in $F$)
Does there exists a prime ...
- 15.4k
1
vote
0
answers
39
views
Unramified lifting in abstract class field theory.
I'm trying to understand Neukirch approach to abstract Class Field Theory as developed in chapter IV of his book "Algebraic Number Theory". I got stuck on the following question: suppose $L/...
0
votes
1
answer
40
views
Kummer extension corresponding to a cohomology class
Let $F$ be a (CM) number field, $k$ be a finite extension of $\mathbb F_p$, $F_S$ be the maximal algebraic extension of $F$ unramified outside $S$. In the 10 author paper p.151, the following are ...
- 445
0
votes
2
answers
75
views
What is the exact definition of what it means for a prime to "split."
I am unsure if the meaning for the term "splits" is standardized. Given a prime ideal in a number ring and an extension of that number ring. Does the prime split if it is not inert or does ...
- 117
0
votes
0
answers
25
views
Irreducibility of modules endowed with a semilinear action
Please bear with me as this will likely be a triviality to many of you - I am probably just confused and would appreciate even an indicative short comment answer.
Setup: let $F$ be a field, and ...
1
vote
3
answers
78
views
Divisibility of discriminants in field towers
I'm studying Marcus's Number Fields, and am stuck in Exercise 8 of Chapter 2, which asks to prove that the $p$-th cyclotomic field contains either $\sqrt{p}$ or $\sqrt{-p}$ depending on whether $p$ is ...
- 2,504
1
vote
0
answers
53
views
What is the Galois group of a polynomial over $\mathbb{F}_p$?
Suppose we have a degree-$n$ polynomial $p(X) \in \mathbb{F}_p[X]$. How does one compute the Galois group of its splitting field, if that's tractable? My understanding so far:
First, we can without ...
- 202
2
votes
1
answer
77
views
Question on Principal Ideals
I've recently begun with Frazer Jarvis' Algebraic Number Theory, and on page 88 (pdf) of the text there's this exercise:
Let R = $\mathbb{Z}[\sqrt{-5}]$, and consider the ideal $\mathfrak{a} = \langle ...
- 23
0
votes
1
answer
72
views
Understanding the proof of the finiteness of the class number
The proof is in Chapter 12 of Rosen and Ireland's A Classical Introduction To Modern Number Theory. Proposition 12.2.3 states that $D/A$ is finite, where $D$ is the ring of all algebraic integers in a ...
- 504
1
vote
0
answers
37
views
Prove that the ring of algebraic integers in an algebraic number field F is integrally closed
The question is posted here but I am still confused. I think the first answer was a wrong argument but I'm not pretty sure.
First, $\alpha\in F$, not $D$, so the elements of matrix $M$ is in $\mathbb{...
- 504
1
vote
1
answer
48
views
Question on principal unit groups in local fields
This is a (likely) simple question, but for some reason I can't see it right now.
In Childress's Class Field Theory book, Lemma 2.5, the author claims that $$(1 + \mathfrak{p}_v^t)^n = 1 + \mathfrak{p}...
- 2,170
3
votes
1
answer
69
views
A property of tamely ramified extensions
Suppose that $L/K$ is a tamely ramified extension of complete fields.
By computing some examples (quadratic fields, and cyclotomic fields $\mathbb Q_p(\zeta_p)/\mathbb Q_p$) I noticed that
$$
v(\...
- 375
4
votes
0
answers
96
views
Ring of integers of "infinite degree" that is a UFD
Given an algebraic extension $K/\mathbb Q$ define the ring of integers $\mathcal O_K$ of $K$ as the integral closure of $\mathbb Z$ inside $K$. My question is the following: If $K/\mathbb Q$ is ...
- 335
0
votes
1
answer
27
views
Question about monogenic extension.
Let $A$ be a DVR with fraction field $K$ (let's say it's a global field for simplicity). Let $L/K$ be a finite (separable) extension, and let $B$ be the integral closure of $A$ in $L$. Must $B=A[\beta]...
- 755
0
votes
0
answers
36
views
Character of Galois group factors through a specific number field
Let $k$ be a number field and $\chi: C_k \rightarrow \{\pm 1\}$ a quadratic idele class character. We know that this corresponds to a character $\rho: \mathrm{Gal}(k^{\mathrm{ab}}/k) \rightarrow \...
0
votes
1
answer
24
views
Norm of an Element depends upon the field extension as well?
Let $K$ be a finite field extension of $\mathbb{Q}$, then I am wondering about properties of $N( \alpha)$, where $N$ denotes the norm of $\alpha$ with respect to the extension of fields $ \mathbb{Q} \...
- 403
9
votes
1
answer
165
views
What is going on with the discriminant of $f^{\circ n}(X)-X$?
Recently, I found this amazing answer of @Mercio where they point out the following about the discriminant of the polynomial $P(X)=\frac{f^{\circ 3}(X)-X}{f(X)-X}$ where $f(X)=X^2+c$.
We can compute ...
- 500
4
votes
1
answer
107
views
Is the linear operator given by multiplication by an element of an algebraic field extension always diagonalizable?
I noticed this to be true for nontrivial extensions given by adjoining a square root. (By diagonalizable i mean over an algebraic closure). Given a nontrivial field extension of form $k\hookrightarrow ...
- 1,053
2
votes
1
answer
95
views
Napkin exercise 54.2.7
Let $K$ a number field and $x\in K$. Then exists $n$ such that $nx \in \mathcal O_K$. The exercise asks us to prove this, and gives two different suggestions about how to do it. One of them is by ...
- 1,053
0
votes
1
answer
42
views
Coprimality of $\frac{(N \mathfrak{b})}{\mathfrak{b}}$ and $\frac{(N \mathfrak{c})}{\mathfrak{c}}$ when $\mathfrak{b}$ and $\mathfrak{c}$ are coprime.
Assume that $\mathfrak{b}$ is an integral ideal of ring of integers of a number field $K$ of $\mathfrak{o}_K$, then $N \mathfrak{b} \in \mathfrak{b}$ follows from lagrange's theorem. Hence $(N \...
- 403
0
votes
0
answers
34
views
Definition of the corresponding valuation from nonzero prime ideals and associated question
$\divideontimes$ Refer to the Janusz, Algebraic number fields, Chap II- Example 2 or Nuekirch, Algebraic number theory, p.69.
Let $\mathcal{o}$ be a Dedekind domain wiht fraction field $K$. Let $\...
- 2,774
0
votes
1
answer
117
views
Fundamental domain of $\Gamma_1(p)$
I am learning about modular curves, and would like more example of fundamental domains of congruence subgroups. Can someone give me the answer to the following question and also give me some pointer ...
- 5,223
1
vote
2
answers
94
views
For Galois extension of number fields, $\mathfrak{P}_j, \mathfrak{P}_k | \mathfrak{p}$ $\Rightarrow$ $L_{\mathfrak{P}_j} \cong L_{\mathfrak{P}_k}$?
Let $L|K$ be a (finite) algebraic Galois extension of number fields. Let $\mathfrak{p} \subseteq\mathcal{O}_K$ be a prime ideal of its ring of integers. Let $\mathfrak{P}_j , \mathfrak{P}_k \subseteq \...
- 2,774
0
votes
0
answers
36
views
Understanding proof of existence theorem of global class field theory more step by step ( Neukirch, Algebraic number theory VI - (6.1) theorem )
I am reading the Neukirch, Algebraic number theory, p.395, proof of (VI)- (6.1) theorem and stuck at some statements :
First, we equip the idele class group $C_K$ with its natural topology ( C.f. his ...
- 2,774
0
votes
1
answer
35
views
Group generated by Image of Uniformizer under local Artin Map
Let $L_w/K_v$ be a finite abelian extension of complete non-archim valued local fields and $\Phi: K_v^{\ast} \to Gal(L_w/K_v)$ canon induced from local Artin map (composed with qoution on codomain ...
- 8,181
2
votes
1
answer
53
views
Field Extension of Local Fields of Degree $d$ containing $d$th root of unity
Let $L_w/K_v$ a finite degree $d=[L_w:K_v]$ Galois extension of $p$-adic fields (ie $L_w,K_v$ are finite extensions of $\Bbb Q_p$).
Assume moreover that $d $ is coprime to $p^{\kappa}-1$, where $p^{\...
- 8,181
1
vote
1
answer
78
views
Roots of a polynomial over $\mathbb{Q}(\sqrt{2},\sqrt{3})$
I am trying to compute the eigenvalues of the unitary DFT of the symmetric group algebra over $\mathbb{Q}$ where a minimal number of square roots have been adjoined. For $n=3$, $K=\mathbb{Q}(\sqrt{2},\...
- 1,545
2
votes
1
answer
133
views
Why isotropic pairing $〈, 〉$ induces $B\cong B^*\stackrel{f^*}{\to}A^*$ and $B\to \text{Coker}f \hookrightarrow A^* $ coincides as a map
Let $A,B$ be abelian groups. $A^*$ be its dual, that is, $A^*=\text{Hom}(A,\Bbb{Q}/\Bbb{Z})$. Suppose there is a non-degenerate pairing $〈,〉: B\times B \to \Bbb{Q}/\Bbb{Z}$ and this pairing induces an ...
- 6,441
1
vote
1
answer
199
views
Does $\alpha-\alpha^r \in K$ imply $\alpha \in K$?
Let $K$ be a finite extension of the $p$-adic number field $\mathbb{Q}_p$.
Assume two algebraic numbers $\alpha-\alpha^r \in K$ and $p\beta-\beta^{p^r} \in K$, where $r \in \mathbb{N}$ and $\alpha,~\...
- 59