Questions related to the algebraic structure of algebraic integers

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0answers
11 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 $\mathcal O_K$?
4
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1answer
25 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 ...
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0answers
27 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 ...
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4answers
156 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 ...
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0answers
46 views

Meromorphic functions on $Y^2 = X^3 + 1$, genus.

Let $k$ be a field of characteristic $\neq 2$, and consider the quadratic extension $F$ of $k(X)$ generated by $\sqrt{X^3 + 1}$. What is/how do I find the genus of $F$? The progress I have so far: ...
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2answers
46 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 ...
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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!
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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 ...
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0answers
33 views
+50

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 ...
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0answers
66 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 ...
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1answer
44 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 ...
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1answer
53 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 ...
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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
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2answers
47 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 ...
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2answers
219 views
+50

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. ...
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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 + ...
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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 ...
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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
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1answer
31 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 $\alpha\in\mathbb Z_p^{\times}$ ...
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1answer
20 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 ...
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0answers
18 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 ...
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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 ...
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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 ...
1
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1answer
30 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
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1answer
32 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 ...
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1answer
30 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 ...
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0answers
29 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 ...
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0answers
27 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 ...
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1answer
47 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 ...
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3answers
82 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
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1answer
34 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
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1answer
38 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 ...
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0answers
47 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 ...
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0answers
30 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 ...
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0answers
66 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
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1answer
48 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 ...
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2answers
15 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 ...
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0answers
21 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 ...
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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 ...
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0answers
31 views

How to show an ideal is principal

Is there a general procedure to check whether or not a prime ideal of the ring of integers $O_K$ is principal. In my case $K$ is a quadratic field, i.e $\mathbb{Q}(\sqrt {d})$, with $d$ square-free.
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2answers
50 views

Determining whether a given algebraic number is an algebraic integer

Let $K$ be a number field and let $\mathcal{O}_{K}$ be the ring of integers of $K$, then given a random element $\alpha$ from our number field $K$, is there any quick and efficient way to determine ...
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1answer
79 views

30th problem of the fifth book of Diophantus;

Is there a complete answer to this problem? I have found Saunderson's answer, but I believe it is missing a few answers. The problem states: $a^2+b^2=d^2 \\ a^2+c^2=e^2 \\ b^2+c^2=f^2$ Saunderson ...
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0answers
21 views

Ring of Integers $\mathcal{O}_{K}$

Let $\{b_{1},b_{2},\dots,b_{d}\}$ be a basis of a Number field $K$. Then if the $b_{i}$ are in $\mathcal{O}_{K}$, the discriminant of $K$ relative to $\{b_{1},b_{2},\dots,b_{d}\}$ is an integer. In ...
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1answer
18 views

Algebraic Number Theory - Rings of Integers

Let $K$ be a number field. Then the ring of integers of $K$ is defined as $K \bigcap \mathbb{B} = \mathcal{O}_{K}$. An integral basis of $K$ is defined as a set of elements $b_{1}, b_{2},\dots, b_{n}$ ...
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3answers
51 views

Cubic root of unity

Is there anyway to solve this without substituting with the values? Prove that: $$\frac{1+10w^2}{1-2w} + \frac{2+17w}{2+3w} = 6$$. (Where $w$ & $w^2$ are the cubic roots of unity)
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1answer
103 views

$CL(O_S) \cong \mathbb{Z}/3\mathbb{Z}$.

Let $F = \mathbb{Q}(T)$ and let $X$ be the set of all places of $F$, and let $S = \{w\} \subset X$ where $w$ is the place of $F$ corresponding to the maximal ideal $(T^3 - 2)$ of $\mathbb{Q}[T]$. Let ...
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1answer
24 views

Why $\mathfrak p_2\cdots\mathfrak p_r\not\subset (a)\mathcal O$

If $(a)\mathcal O\subset\mathfrak p_1$ and $r$ is the minimal number such that $\mathfrak p_1\cdot\mathfrak p_2\cdots\mathfrak p_r\subset (a)\mathcal O$ then $\mathfrak p_2\cdots\mathfrak ...
3
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0answers
30 views

Determine the ideal class group of $K = \mathbb{Q}(\sqrt{-78})$

The question is to determine the ideal class groups of $K = \mathbb{Q}(\sqrt{-78})$. Well, by the Minkowski bound, we just need to check for ideals such that: $N(I) \leq \dfrac{4\sqrt{-78}}{\pi} ...
5
votes
3answers
77 views

is a number field by definition a subfield of $ \mathbb C $?

I have seen that some authors are defing the number field as a subfield of $ \mathbb C$ which is a finite extension of the rational numbers $ \mathbb Q $, while some others without referering to ...
2
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2answers
23 views

behavior of a rational prime in quadratic extension (definition)

Let $ \mathbb Q \subset K=\mathbb Q (\sqrt{-n}) \subset L $, where $K/ \mathbb Q $ is a finite extension (i.e. $K$ is a number field) and $L/K$ is a maximal uramified abelian extension. If $p ...