Questions related to the algebraic structure of algebraic integers

learn more… | top users | synonyms

4
votes
1answer
44 views

Field extension $\mathbb{Q}[a]$ with $a,b$ algebraic integers: show $bf'(a)\in\mathbb{Z}[a]$

I am trying to understand a paper which seems to claim the following: Let $f$ be monic irreducible in $\mathbb{Z}[X]$, and $a$ be one of its roots in $\mathbb{C}$. Let $b$ be an algebraic integer in ...
0
votes
0answers
16 views

What is the additive inverse of a 2-adic Witt vector?

The ring of $p$-adic Witt Vector is defined as follows: First, let $X_0, X_1, \cdots$ be an infinite sequence of unknowns and put \begin{equation*} W_n = X_0^{p^n} + pX_1^{p^{n-1}} + \cdots + ...
0
votes
1answer
54 views

$\log (1+x)$ when $x$ is $p$-adic

It's written when $x$ is $p$-adic integer then $\log (1+x) = \sum (-1)^{n-1}\frac{x^n}{n}$ converges, I don't understand what this statement mean. Can one please explain me ?
2
votes
1answer
27 views

Valuation of a particular element

I am tying to compute the valuation of a particular element of $\mathbb{Q}_p$. I am trying to compute $\operatorname{val}_p(P)$ where $P=\frac{\log(1+p^2)}{\log(1+p)}$ and $\log$ is the $p$-adic ...
0
votes
1answer
29 views

Prove that $L(\psi_N , s) = \zeta(s) \prod_{p \mid N}(1 − p^{-s})$

Let $N$ be a positive integer, and let $\psi_N$ be the trivial Dirichlet character with conductor $N,$ so $\psi_N (a) = 0$ if $\gcd(a, N) \ne 1$ and $\psi_N (a) = 1$ if $\gcd(a, N) = 1.$ Prove that ...
3
votes
2answers
66 views

Prove the class number of $\mathbb{Z}[\sqrt{-5}]$ is $2$.

Prove the class number of $\mathbb{Z}[\sqrt{-5}]$ is $2$ . This argument Prove that the class number of $\mathbb{Z}[\zeta_3]$ is $1$ doesn't work since $\mathbb{Z}[\sqrt{-5}]$ is not a unique ...
4
votes
1answer
43 views

Show an intersection of Galois groups is trivial

Let $L/K$ be a finite abelian extension of number fields, and for an extension of places $w/v$ consider the local Artin map $\Phi: K_v^{\ast} \rightarrow Gal(L_w/K_v)$, defined via the global Artin ...
3
votes
2answers
88 views

Transform a polynomial so that positive roots are shifted right and negative roots are shifted left

I'm trying to figure out if it is possible to shift the roots of a polynomial outward, instead of to the left or right. Its relatively simple to shift all the solutions in one direction by ...
1
vote
0answers
37 views

Is this theorem in Samuel's Algebraic theory of numbers true?

I am not convinced by both the statement of the theorem and the given proof. Here it is: Theorem 1: Let K be a field of characteristic zero or a finite field, let K' be an extension of finite degree n ...
3
votes
1answer
49 views

Weil pairing, cyclotomic field in division field and determinant map for ell curve and abelian variety

Question 1: If $E/K$ is an elliptic curve defined over a number field $K$, then the Weil Pairing gives me that $K(\mu_n) \subseteq K(E[n](\bar{K}))$. If i identify $Gal(K(\mu_n)/K)$ with a subgroup ...
0
votes
0answers
172 views

“Necessary” condition for Power Diophantine Equation.

Motivation: Brocard’s problem $n!+1$ being a perfect square Observations: Given a power Diophantine equation of $k$ variables with a “general solution” (provides infinite integer solutions) to ...
2
votes
0answers
25 views

Irreducible polynomials and poving the ring of intergers is a PID

My question isn't too hard I think I'm just a little stumped on how to tackle the second part. $ Let \ K=\Bbb Q(\alpha)$ where $\alpha$ is a root of $f(x)=x^3+2x+1$ 1) Show that $f(x)$ is ...
1
vote
1answer
27 views

Analytic formulas for special values of $L$-functions

In The Princeton Companion to Mathematics, IV.1, “Algebraic Numbers”, the conditionally convergent series \begin{equation} ...
2
votes
1answer
56 views

Difference between the Artin symbol and the Frobenius element?

The title says it all really! I have been reading 'Primes of the form $x^2+ny^2$', by Cox, and in chapter 5 he introduces the Artin Symbol, which for a field extension, $L/K$ is the unique element, ...
1
vote
1answer
28 views

Rank of non-zero integral ideal as a module

I am reading Pierre Samuel Algebraic Theory of Numbers. On pages 57, Let $K$ be a number field and let $n$ be its degree. $\sigma$ is the canonical imbedding of $K$ in $\mathbf R^{r_1} \times ...
9
votes
3answers
77 views

Must an irreducible element in $\mathbb{Z}[\sqrt{D}]$ have a prime norm?

Let $D\in \mathbb{Z}$ where $D$ is not a perfect square. Prove that if $\alpha\in \mathbb{Z}[\sqrt{D}]$, and $\alpha= a+b\sqrt{D}$ with: $|a^2-Db^2| = p$, a rational prime, ...
4
votes
0answers
55 views

Matrix which represents the product of ideal classes of 2 matrices.

Let $f(x)$ be an irreducible monic degree $n$ polynomial with $\mathbb{Z}$-coefficients and $\Theta$ be a root of $f(x)$. There is an old theorem of Latimer and MacDuffee that there is a 1-1 ...
3
votes
2answers
70 views

What is the class group of $\Bbb{Q}(\sqrt{-41})$?

What is the class group of $\Bbb{Q}(\sqrt{-41})$? I've found that it's generated by $P_2, P_3, P_5, P_7$ as per Dedekind's theorem, but I'm having a bit of trouble finding the relations between the ...
2
votes
1answer
24 views

Archimedean completion of a number field

Let $K$ be a number field. I want to show that Every Archimedean absolute value of $K$ is equivalent to the absolute value $|x|:=|\sigma(x)|_\infty$ for $x\in K$ where $\sigma$ is an embedding of ...
3
votes
0answers
59 views

Archimedean places of a number field

Let $K$ be a number field with an Archimedean absolute value $|\cdot |$ and let $\bar{K}$ be the completion of $K$ wrt this valuation. Then $\bar{K}\cong \mathbb R $ or $\mathbb C$. My question is: ...
0
votes
1answer
51 views

Learning Algebraic Number theory

I am looking for a good reference to self study algebraic number theory, as no undergraduate course is given at the university. I've web-searched a lot of online notes and courses, and I don't seem to ...
2
votes
1answer
39 views

Hasse invariant of quaternions over $\mathbb{Q}_p$

I am trying to compute the Hasse invariant of the quaternion algebra over $\mathbb{Q}_p$. I denote this algebra by $H$, and I'm assuming $p\equiv 3\pmod{4}$. So, $\mathbb{Q}_p(i)$ is an unramified ...
1
vote
0answers
36 views

Who first used the notation $\mathcal{O}_K$ for ring of integers?

I think this is a standard notation since almost every author uses it, but who came up with the notation? After all, what does $\mathcal{O}$ in $\mathcal{O}_K$ stand for? Thanks in advance.
1
vote
0answers
28 views

Maximal p-subgroup of inertia group.

We know from the theory that if $\mathbb{L}$ is a finite Galois extension of the local field $\mathbb{K}$ then the ramification group $G_1$ is a $p$-group where $p$ is the characteristic of the ...
2
votes
5answers
107 views

Proving that $\cos(2\pi/n)$ is algebraic

I want to prove this without using any of the properties about the field of algebraic numbers (specifically that it is one). Essentially I just want to find a polynomial for which $\cos\frac{2\pi}{n}$ ...
1
vote
1answer
54 views

Every finite abelian extension of Q contains a totally real subfield of index 2?

I can reduce this to the case of cyclotomic field extensions, by embedding the abelian extension into a cyclotomic extension and using the "sliding-up" lemma. I am stuck on how to prove this for the ...
2
votes
1answer
47 views

Lower bounds on the index of $\mathbf Z[X]/(P)$ in the ring of integers of a number field

Let $P$ in $\mathbf Z[X]$ be an irreducible polynomial. Let $\mathcal O$ be the ring of integers of the number field $K:=\mathbf Q[X]/(P)$ and $i$ be the index of $\mathbf Z[X]/(P)$ in $\mathcal O$. ...
8
votes
3answers
96 views

Integers of the form $x^2+2y^2$.

I'm stuck in the following problem: prove an integer $n$ is of the form $x^2+2y^2$ if and only if every prime divisor $p$ of $n$ that is congruent to $5$ or $7\bmod8$ appears with an even exponent. I ...
6
votes
1answer
47 views

What does it mean for a prime ideal to split completely?

See here. What does it mean for a prime ideal to split completely?
5
votes
1answer
70 views

Extension $\mathbb{Q}(\sqrt{-5}, i)/\mathbb{Q}(\sqrt{-5})$, splitting.

In the extension $\mathbb{Q}(\sqrt{-5}, i)/\mathbb{Q}(\sqrt{-5})$, must principal prime ideals of $\mathbb{Z}[\sqrt{-5}]$ necessarily split into 2? Must nonprincipal prime ideals not split? ...
5
votes
0answers
48 views

An infinite prime can ramify right? (So what is Neukirch talking about?)

I have been under the impression for several years that if $L/K$ is an extension of number fields, then an infinite place of $K$ is said to ramify in $L$ if it comes from a real embedding of $K$ which ...
1
vote
0answers
40 views

Geometric structure on the set of valuation rings of a field

Let $K$ be a field. Let $\mathcal{O}_K$ be the intersection of all valuation rings with quotient field $K$. Can someone give an example of a field $K$ in which we don't have a bijection of sets: $$ ...
1
vote
1answer
77 views

How to prove that the ring of algebraic integers is a Bézout domain?

I was told that the ring of all algebraic integers (that is, the complex numbers which are roots of a monic polynomials with integral coefficients) is a Bézout domain. But I have no idea how to prove ...
3
votes
1answer
41 views

Inertia group modulo $Q^2$

Let $L/K$ be a normal number field extension with ring of integers $\mathcal O_L/\mathcal O_K$. Let $Q$ be a prime ideal of $\mathcal O_L$ and inertia group $E = \{g \in ...
6
votes
2answers
82 views

How does one prove that $\mathbb{Z}\left[\frac{1 + \sqrt{-43}}{2}\right]$ is a unique factorization domain.

By extending the Euclidean algorithm one can show that $\mathbb{Z}[i]$ has unique factorization. This logic extends to show $\mathbb{Z}\left[\frac{1 + \sqrt{-3}}{2}\right]$, $\mathbb{Z}\left[\frac{1 ...
3
votes
0answers
42 views

Can we find the 18 imaginary quadratic ffields with class number 2 algorithmically?

I am reading about the class number problem. There is a well known complete list of imaginary quadratic fields $\mathbb{Q}(\sqrt{-d})$ with class number $1$. I found a paper by Stark that says ...
4
votes
4answers
69 views

Product of two integers of the form $x^2+my^2$ is of the same form.

Let $x,y,a,b\in \mathbb Z$. Prove that there are integers $c$ and $d$ so that \begin{equation*} (x^2+y^2m)(a^2+b^2m)=c^2+d^2m. \end{equation*} I'm stuck, I took the product and got ...
3
votes
2answers
84 views

Elliptic curve $y^2= x^3 + x$ over the finite field $\mathbb{F}_p$ with $p \geq 3$.

Consider the elliptic curve $$E: y^2= x^3 + x$$ over the finite field $\mathbb{F}_p$ with $p \geq 3$. I want to show that $|E(\mathbb{F}_p)| \equiv 0 \mod 4$. I know that, if $p \equiv 3\mod 4$, ...
3
votes
1answer
32 views

Is $\mathcal O^{\ast}/U^{(n)}\cong (\mathcal O/\mathfrak p)^{\ast}$ as topological groups?

I have the following: $K$ is a field with discrete valuation $v$, $\mathcal O$ its valuation ring and $\mathfrak p$ the maximal ideal and $U^{(n)}=1+\mathfrak p^n$ the $n$-th unit group for $n\geq 1$. ...
1
vote
1answer
21 views

Decomposing the ring of integers of a number field as a sum of prime plus field fixed by ramification group

Let $L/K$ be a normal number field extension with ring of integers $\mathcal O_L/\mathcal O_K$. Let $Q$ be a prime ideal of $\mathcal O_L$ and inertia group $E = \{g \in ...
3
votes
0answers
44 views

Order of prime ideals over split primes in the class group.

Let $P$ be a prime ideal of $\mathcal O_K$ ($K$ a quadratic field) and let $P$ have norm $p$ where $p$ is a split prime. Is it possible for the ideal class $[P]$ to have order less than three? I feel ...
3
votes
1answer
96 views

The elliptic curve $y^2 = x^3 + 2015x - 2015$ over $\mathbb{Q}$

Consider the elliptic curve \begin{equation*} E: y^2 = x^3 + 2015x - 2015~\text{over}~\mathbb{Q}. \end{equation*} I want to prove that $|E(\mathbb{F}_7)| = 12$, that $|E(\mathbb{F}_{19})| = 19$ and ...
20
votes
1answer
338 views

Serge Lang never explains anything

On page 149 of Algebraic Number Theory by Serge Lang, I'm trying to understand why the inclusion $$k^{\ast}N_k^K(J_K) \cap J_c \subseteq \psi^{-1}(P_c \mathfrak N(c))$$ is true. I've been trying for ...
4
votes
1answer
38 views

Which properties determine the uniqueness of the local Artin map?

Any abelian extension of local fields can be realized as the completion of a global abelian extension. So let $L/K$ be abelian, $w/v$ an extension of places. From the global Artin map on ideles we ...
2
votes
0answers
63 views

Is there a classification of ideals of $\mathcal O_K$ ($K$ quadratic) over ramified and split primes depending on $d \pmod 4$?

I am unsure if the following argument is correct. I have not seen something like this in my course, so I'm a bit skeptical, since this seems like a very simple way of computing norms of ideals. If ...
2
votes
0answers
22 views

Why is $m \infty$ the conductor of $K = \mathbb{Q}(\zeta_m)/\mathbb{Q}$?

Wouldn't this be saying that for all $p$ dividing $m$, $1 + p^{\operatorname{ord}_p(m)} \mathbb{Z}_p$ is contained in the group of local norms $N_{\mathfrak p/p}(K_{\mathfrak p})$, where $\mathfrak p$ ...
2
votes
0answers
23 views

Show that $U_i \rightarrow U_{i+e}, x \mapsto x^p$ is an isomorphism

Show that $U_i \rightarrow U_{i+e}, x \mapsto x^p$ is an isomorphism? Let $K$ be a finite extension of $\mathbb{Q}_p$ with uniformizer $\pi$, prime $\mathfrak p$, and ramification index $e = ...
4
votes
2answers
81 views

How to prove that any infinite algebraic extension of a complete field is never complete?

My first idea is using Baire category theorem since I thought an infinite algebraic extension should be of countable degree. However, this is wrong, according to this post. This approach may still ...
4
votes
0answers
82 views

$\mathbb{Q}$ isn't a number field for SAGE

This is more a question about the weird behavior of SAGE: ...
2
votes
4answers
63 views

Determining whether or not an element is integral over $\mathbb Z$

I want to prove that $\alpha=\frac{\sqrt[3]{2}}{2}$ is not integral over $\mathbb{Z}$ (i.e. not an algebraic integer). Is there a way to modify the following reasoning to make it work? A quick check ...