Questions related to the algebraic structure of algebraic integers

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2
votes
1answer
25 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 $\mathcal O$. ...
4
votes
2answers
58 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 ...
5
votes
1answer
37 views
5
votes
1answer
57 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? ...
3
votes
0answers
29 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
27 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
54 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
34 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 ...
4
votes
2answers
61 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
33 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
66 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
67 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
0answers
22 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
20 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
37 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
73 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 ...
7
votes
0answers
79 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 ...
3
votes
0answers
14 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
51 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 ...
1
vote
0answers
10 views

Intersection of decomposition groups in abelian extensions

Let $L/K$ be a normal number field extension such that $G = \operatorname{Gal}(L/K)$ is abelian. Further let $P$ be a prime in $\mathcal O_K$ and $Q_1,Q_2$ distinct primes of $\mathcal O_L$ that lie ...
2
votes
0answers
15 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
19 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 = ...
2
votes
0answers
22 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
70 views

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

This is more a question about the weird behavior of SAGE: ...
2
votes
4answers
59 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 ...
1
vote
2answers
55 views

Can a unit of infinite order in algebraic integers of a number field be an arbitrarily high power of another unit?

Is there a number field $K$ and a unit $u \in \mathcal{O}_K^{\times}$ of infinite order which can be written as an arbitrarily high power of another unit? I think the answer is no, because a ...
0
votes
2answers
33 views

How do I compute the norm of a non-principal ideal of the ring of integers of a quadratic field without using ''large'' results

I am trying to compute the norm of the ideal $I=(7, 1+\sqrt{15}) \trianglelefteq \mathbb Z[\sqrt{15}],$ the ring of integers of $\mathbb Q[\sqrt{15}].$ I knew $I^2$ would be principal, as $I\bar ...
1
vote
0answers
20 views

Why are inertia and decomposition groups only defined over normal extensions?

Can't we define them as the subgroups of the automorphism group of an arbitrary extension $L/K$ that fixes a prime $Q$ in $\mathcal O_L$ over a prime $P$ in $\mathcal O_K$? An ideal answer would ...
4
votes
1answer
29 views

Visualizing Euclidean Algorithm in $\mathbb{Q}(\sqrt{-7})$ and $\mathbb{Q}(\sqrt{-11})$ with Convex Geometry

In an attempt to answer one of the bounty questions, I have started picturing Euclidean division in quadratic fields. In theory we would like the equation: $$ a = b\,q + r ...
1
vote
0answers
34 views

A question about a property of Gauss sum.

I am reading the book and I have some questions about Gauss sum. The Gauss sum is defined in the end of page 4, formula (1.14), by \begin{align} g(m,c)=\sum_{a \mod c} \left( \frac{a}{c} \right)_n ...
13
votes
5answers
190 views

Is 'Algebraic Number Theory' the study of the theory of algebraic numbers, or is it the study of the theory of numbers from an algebraic viewpoint?

Is Algebraic Number Theory the study of the theory of algebraic numbers? Or is it the study of the theory of numbers from an algebraic viewpoint? Or is it both? I know I can just find a wiki article ...
10
votes
0answers
129 views
+100

Meromorphic functions on $U^2 = T^3 + 1$, genus.

Let $k$ be a field of characteristic $\neq 2$, and consider the quadratic extension $F$ of $k(T)$ generated by $\sqrt{T^3 + 1}$. What is/how do I find the genus of $F$? The progress I have so far: ...
0
votes
2answers
49 views

Extension of prime ideals in Dedekind domains

In various textbooks and lecture notes on algebraic number theory, I have found the following claim without proof: Let $R$ be a Dedekind domain with field of fractions $F$ and let $S$ be its integral ...
2
votes
0answers
12 views

Example of indecomposable ideal which is not prime

everybody! Can someone give me an example of an ideal which is indecomposable, but not a prime ideal of some ordering $O$ in quadratic field? Thanks!
0
votes
0answers
3 views

Is any F-stable maximal torus contained in some F-stable maximal Borel subgroup?

Denote by $\mathbb{F}_q$ the finite field with $q$ elements, and denote by $\bar{\mathbb{F}}_q$ its algebraic closure. Let $G$ be an affine algebraic group over $\bar{\mathbb{F}}_q$, and let $F$ be a ...
7
votes
0answers
92 views

What does the ideal norm of matrix elements really mean?

Say we have a number field $K$ (specifically, an imaginary quadratic field) and a $2\times2$ matrix $\sigma=\pmatrix{a&c\\b&d}$ with elements $a,b,c,d\in\mathcal O_k$, the ring of integers of ...
1
vote
0answers
72 views

Solving $|x|^2=\sqrt{2}-1$ in $\mathbb{Z}[\xi_8] $

Is there a solution of the equation $|x|^2=\sqrt{2}-1$ in $\mathbb{Z}[\xi_8]$, where | | means the complex absolute value? In general, can I solve the equations of the form $|x|^2=c$ in each ring of ...
0
votes
1answer
48 views

Order of element in algebraic group

Denote by $\mathbb{F}_q$ the finite field with $q$ elements, and denote by $\bar{\mathbb{F}}_q$ its algebraic closure. Let $G$ be an affine algebraic group over $\bar{\mathbb{F}}_q$, and let $F$ be a ...
2
votes
1answer
59 views

Class group of $\mathbb Q(\sqrt{-55})$ and finding representatives for ideal classes

My first step in computing the class group of $\mathbb Q(\sqrt{-55})$ was to compute the Minkowski bound. Initially, I said $\lambda(-55)=2\sqrt{-55}/\pi<2(8)/3<6$ and I went the normal way of ...
2
votes
0answers
24 views

Order of elements in algebraic group

Denote by $\mathbb{F}_q$ the finite field with $q$ elements, and denote by $\bar{\mathbb{F}}_q$ its algebraic closure. Let $G$ be an affine algebraic group over $\bar{\mathbb{F}}_q$, and let $F$ be a ...
2
votes
2answers
50 views

Is the minimal number of generators of an ideal the rank of the ideal as a free $\mathbb Z$-module?

In an algebraic number theory course, my lecturer said that any ideal of $\mathcal O_K$, where $K$ is a quadratic number field, is generated by at most two elements. I am wondering why this is. When ...
18
votes
2answers
320 views

Without using Heegner-Stark-Baker, $\mathbb{Q}(\sqrt{-11})$ has class number $1$.

Prove that $\mathbb{Q}(\sqrt{-11})$ is of class number $1$. I have found that the ideal $(2)$ of the integer ring $\mathbb{Z}[(1 + \sqrt{-11})/2]$ of $\mathbb{Q}(\sqrt{-11})$ is a prime ideal. ...
1
vote
1answer
21 views

Finding $\mathbb Z[\zeta_p]^∗$, the group of units of $\mathbb Q(\zeta_p$)

Let $p$ be an odd prime and $\zeta_p$ be a primitive $p$-th root of unity. I'm trying to prove that $\mathbb Z[\zeta_p]^∗$, the group of units of $\mathbb Q(\zeta_p$) is $(\zeta_p)\mathbb Z[\zeta_p + ...
0
votes
0answers
35 views

Roots of unity in $\mathbb Q(\zeta_n)$

If $n$ is a positive integer then the roots of unity in $\mathbb Q(\zeta_n)$, with $\zeta_n=e^{\frac{2\pi i}{n}}$ is a cyclic group and is generated by $\zeta_{\tilde n}$, with ${\tilde ...
7
votes
1answer
47 views

Poles of a sum of functions

The other question is here. Let $F$ be a function field in one variable over a field $k$. Let $S$ a nonempty finite subset of the set of all places of $F$. Prove that if $P \in S$, there is an ...
2
votes
1answer
32 views

Squares in $\mathbb Q_p$ are $p^{2n}\alpha$

If $p$ is an odd prime, then the squares in the field of p-adic numbers $\mathbb Q_p$ are the elements are $0$ or of the form $p^{2n}\alpha$, $n\in\mathbb Z$ and $\alpha\in\mathbb Z_p^{\times}$ ...
0
votes
1answer
22 views

Surjectivity of ring homomophism induced by Frobenius endomorphism

Denote by $F_q$ the finite field with $q$ elements, and denote by $\bar{F_q}$ its algebraic closure. Let $V$ be an affine $\bar{F_q}$-variety and $F$ be the Frobenius endomorphism corresponding to an ...
2
votes
0answers
21 views

Lists of negative discriminants by class group?

Is there a handy listing of the discriminants of imaginary quadratic fields having a given ideal class group? It would be nice to use such a resource as a source of examples. For example, we're all ...
1
vote
2answers
29 views

Conductor of a ring

An easy (possibly trivial) question from Neukirch's Algebraic Number Theory, p.47 Let $A$ be a Dedekind domain, $K$ its fraction field, $L$ a finite separable extension of $K$ and $B$ the integral ...
8
votes
0answers
74 views

Classification of all subrings

Let $R$ be an integral domain whose underlying additive group is finitely generated free and whose field of fractions $K$ is a finite Galois extension of $\mathbb{Q}$. Is there a method of ...