Questions related to the algebraic structure of algebraic integers

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3
votes
1answer
93 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
325 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
37 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 = ...
3
votes
1answer
56 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
80 views

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

This is more a question about the weird behavior of SAGE: ...
2
votes
4answers
62 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
60 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
37 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
22 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
73 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
43 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 ...
14
votes
5answers
227 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 ...
21
votes
0answers
359 views

Meromorphic functions on $U^2 = T^3 + 1$, cokernel of $O_S \to F_\infty/O_\infty$.

Let $k$ be a field of characteristic $\neq 2$, and consider the quadratic extension $F$ of $k(T)$ generated by $\sqrt{T^3 + 1}$. I guess to avoid confusion, I'm asking the new question: what is the ...
0
votes
2answers
60 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
4 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
99 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
75 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
52 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
67 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
26 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 ...
1
vote
2answers
67 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
343 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
25 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
39 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 ...
8
votes
1answer
68 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
36 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
26 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 ...
3
votes
0answers
28 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 ...
2
votes
2answers
47 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
87 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 ...
1
vote
1answer
44 views

Existence of induced map on Divisor Class Group?

Let $f: X \rightarrow Y$ be a morphism of noetherian, integral schemes, regular in codimension 1 (so we can talk about Weil divisors). I am wondering whether there is an induced map on divisor class ...
2
votes
1answer
49 views

Transitivity of the discriminant of number fields

Let $M/L/K$ be a tower of number fields with discriminant of $M/K: d_M$ and of $L/K: d_L$. I would like to find a transitivity theorem for the discriminant and by letting $p_i$ and $q_i$ be integral ...
1
vote
1answer
36 views

Ramification in $\mathbb Q(\zeta_5, \sqrt[5]2)/\mathbb Q(\zeta_5)$

Let $F=\mathbb Q(\zeta_5,\sqrt[5]2)$ and $K=\mathbb Q$ where $\zeta_5$ is a primitive $5$th root of unity and let $p=73$ be a prime in $K$. Fix primes $\mathfrak p$ and $\mathfrak q$ above $73$ in ...
3
votes
1answer
41 views

Every unit in $\mathcal O_K$ is equal to a power of $\zeta$ times a real unit in $\mathcal O_K$

Every unit in $\mathcal O_K$ is equal to a power of $\zeta$ times a real unit in $\mathcal O_K$, with $\zeta:=e^{2\pi\sqrt{-1}/p}$ and $K:=\mathbb Q(\zeta)$ The proof is below, but I don't ...
2
votes
0answers
32 views

Ideal classes of $\mathcal{O}_k$

How can I show if two ideals are in the same ideal class when considering the ideal classes of $\mathcal{O}_k$? Could someone show me an example of discarding ideals that have been used before? Thank ...
1
vote
1answer
53 views

Proving that an element generates $\mathcal{O}_K^*/ (\mathcal{O}_K^*)^3$

Let $K = \mathbb{Q}[x] / \langle x^3 + 2x^2 + 6x + 6\rangle$. The polynomial has a single real root and its discriminant is $-588$. Let $\alpha$ be the image of $x$ in $K$. Then how would I show that ...
3
votes
3answers
95 views

Explicit description of $\Bbb Q_p \cap \bar{\Bbb Q}$

Note that we can embed $\Bbb Q_p$ into $\Bbb C$, as it is discussed here. But as far as I understand, this embedding sends the power series to transcendental elements, so we can't certainly embed ...
2
votes
1answer
35 views

Trouble constructing $\mathbb Z_3[x]/(x^2+1)$

If I have $\mathbb Z_3[x]/(x^2+2x+2)$, I can construct a field by letting $x^2=x+1$. The reps are: $0$ $1$ $x$ $x^2=x+1$ $x^3=x^2+x=x+1+x=2x+1$ $x^4=2x^2+x=2x+2+x=2$ $x^5=2x$ $x^6=2x^2=2x+2$ ...
3
votes
1answer
42 views

Solving Mordell Equations

I am looking at the solution provided in my lecture notes for solving this particular mordell equation: $$y^2 = x^3 − 2$$ which factors into: $$ (y- \sqrt {-2})(y+ \sqrt {-2}) = x^3 $$ In the ...
2
votes
0answers
50 views

Computing the class group of $\mathbb{Z}[\alpha]$, where $\alpha$ is a root of an irreducible polynomial.

As the title says, I am wanting to compute the class group of $\mathbb{Z}[\alpha]$, where $\alpha$ is the real root of the irreducible polynomial $f(x)=x^3+2x^2+6x+6.$ Denoting by $K$, the field ...
0
votes
0answers
37 views

Ideal class groups of a real quadratic field.

I am trying to compute the ideal class group of a real quadratic field of an integer such that it is congruent to $1\pmod 4$ and $1\pmod 8$ and it's Minkowski bound is around 4. The problem is that I ...
8
votes
0answers
81 views

Genus of $k(T)$?

Let $F$ be a function field in one variable with total constant field $k$, let $X$ be the set of all places $F$, and let $S$ be a nonempty finite subset of $X$. Then the genus of $F$ is equal to the ...
2
votes
1answer
61 views

Non-monic polynomial with roots on the unit circle

When setting up to prove Dirichlet's Unit Theorem, we show that all roots of unity in a number field K are algebraic integers. Further, if all conjugates of $x \in \mathcal{O}_K$ have modulus 1 then ...
0
votes
2answers
20 views

Reducing an ideal to an ideal generated by fewer elements.

For $d=-31$, and $I=(2, 1/2 +\sqrt{-31}/2)$ I've been told that $I\cdot\overline{I}=(2)$ I've written $I\cdot\overline{I}= (4, 1-\sqrt{-31} , 1+\sqrt{-31}, 8) $ In what ways am I allowed to reduce ...
1
vote
0answers
24 views

Action of the group ring $\mathbb{Z}[\text{Gal}(K/\mathbb{Q})]$ on the field $K$

Let $K$ be an algebraic number field, let $G$=Gal($K/\mathbb{Q}$). Let $\mathbb{Z}[G]$ be the group ring, or the set of formal sums $$\left\lbrace\sum a_i\sigma_i : a_i\in \mathbb{Z}, \sigma_i \in ...
1
vote
0answers
22 views

Trouble finding the norm of the two following ideals

Given that $\alpha$ is the root of the polynomial, $x^3 - x - 1$ is $\alpha$ and $K=\mathbb{Q}(\alpha)$, show that the norm of the ideal $\langle 5, \alpha-2\rangle$ is $5$ and the norm of the ideal ...