Questions related to the algebraic structure of algebraic integers

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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
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 ...
2
votes
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 ...
0
votes
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 ...
8
votes
0answers
67 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
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 ...
0
votes
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 ...
1
vote
0answers
22 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
17 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 ...
1
vote
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.
1
vote
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 ...
4
votes
1answer
80 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 ...
0
votes
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 ...
0
votes
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 $b_{1}, b_{2},\dots, b_{n}$ ...
1
vote
3answers
52 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)
12
votes
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 ...
2
votes
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
votes
0answers
31 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
votes
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 ...
5
votes
2answers
56 views

Solving polynomial equations using more than radicals

It is well known that one cannot solve every polynomial equation over $\Bbb Q$ using just radicals. In other words, let $A_n = \{x^n - a\mid a \in \Bbb Q\}$, $A = \cup_n A_n$ and $\bar{\Bbb Q}$ the ...
0
votes
1answer
21 views

For fractional ideal $I$ why is $I\cap R \supsetneq \{0\}$?

In the proof in my textbook that a fractional ideal $I$ in a quotient field $K$ of an integral domain $R$ has an inverse $$I^{-1} = \{ x\in K : x I \subseteq R\}\,,$$ it is used that there exists an ...
1
vote
0answers
23 views

Simple examples of fractional ideals

Let $K$ be the quotient field of an integral domain $R$. A fractional ideal $I$ is a subset of $K$ not $\{0\}$, for which a $0 \neq r \in R$ exists so that $r I \subseteq R$ is an ideal in $R$. ...
1
vote
0answers
23 views

Understanding linearly disjoint fields and how their rings of integers interact in a proof

Let $L,L'$ be linearly disjoint number fields (i.e. finite-degree extension of $\mathbb{Q}$). Their rings of integers are denoted $O_L,O_{L'}$. I am trying to understand a proof of how if $p$ is ...
4
votes
1answer
59 views

The splitting of an ideal

Let $K = \mathbb{Q}(\sqrt{-5})$. Now the ring of integers $\mathcal{O}_{K}$ is $\mathbb{Z}[i\sqrt{5}]$. I want to describe the ideal $(2)$ in $\mathbb{Z}[i\sqrt{5}]$ using the prime factorization. ...
4
votes
5answers
74 views

Show that $\dfrac{m - \sqrt[3]{2}}{\sqrt[3]{3}}$ is an algebraic integer.

Let $m$ be an integer such that $m \equiv 2 \pmod 3$. Show that the number $$\dfrac{m - \sqrt[3]{2}}{\sqrt[3]{3}}$$ is an algebraic integer. The usual technique, doing $x = \dfrac{m - ...
10
votes
3answers
148 views

Something screwy going on in $\mathbb Z[\sqrt{51}]$

In $\mathbb Z[\sqrt{6}]$, I can readily find that $(-1)(2 - \sqrt{6})(2 + \sqrt{6}) = 2$ and $(3 - \sqrt{6})(3 + \sqrt{6}) = 3$. It looks strange but it checks out. But when I try the same thing for ...
1
vote
1answer
77 views

The ideal $(p)$ always factors in the ring of integers of $\mathbb Q(\sqrt{2},\sqrt{3})$

Let $p$ be a prime integer. Is there a relatively elementary way to see that $(p)$ is never prime in the ring of integers of $\mathbb Q(\sqrt{2},\sqrt{3})$? One can prove this by looking at the ...
2
votes
0answers
27 views

The norm of an ideal and the norms of its elements

Let $F$ be a number field, $\mathfrak a$ a fractional ideal. If $\mathfrak a$ is a prime ideal in $\mathcal O_F$ lying over prime ideal $p\mathbb Z$ in $\mathbb Z$ then define its norm as $p^f$ where ...
0
votes
1answer
61 views

Prerequisites for Teichmuller Theory

What kind of prerequisites would be required for [Inter-Universal] Teichmuller theory or at least the closest generally known area near Mochizuki's work? (Starting from undergraduate math). I'd ...
1
vote
0answers
10 views

Why is the separable closure of a field in it's completion not complete.

I am currently reading "Algebraic Number Theory" By Neukirch and I am a bit confused by what is here. It is on page 143, it states this, $(K,v)$ is a nonarchimedian valued field and ...
1
vote
0answers
41 views

Show $\text{Gal}(K_\infty/\mathbb Q)\cong \mathbb Z_p^{\times}$

Let $\zeta_{p^n}$ be the primitive $p^n$-th root of unity where $p$ is a prime and $K_n=\mathbb Q(\zeta_{p^n})$ the $p^n$-th cyclotomic field. Let $K_\infty=\bigcup K_n$. Could someone give a proof ...
1
vote
3answers
39 views

Is it possible to factor $27$ in $\mathbb Z[\sqrt7]$ other than in $27=3\cdot3\cdot 3$?

I need to know if there exists an element of norm$27$ in $\mathbb Z[\sqrt 7]$, that is, whether there are any integer solutions to $a^2 - 7b^2 = 27$. When I used modular arithmetic, I find out that ...
0
votes
0answers
14 views

Volume of the lattice generated by an ideal

Let $F$ be a totally real number field, $\mathfrak a \subset F$ a fractional ideal. Consider a lattice in $\mathbb R^n$ consisting of vectors $(\sigma_1(v),..\sigma_n(v))$, where $\sigma_1,..\sigma_n$ ...
0
votes
1answer
43 views

field of rational functions of a curve

Let $C$ be the algebraic curve defined by the modular polynomial $\phi_N$ of order $N>1$ over the rational numbers, i.e. \begin{equation}C:=\text{specm}(\mathbb{Q}[X,Y]/\phi_N(X,Y)). \end{equation} ...
0
votes
1answer
36 views

Artin reciprocity theorem for Hilbert class field

In Cox's book "Primes of the form $x^2 + ny^2 $..." gives the following statement of Artin reciprocity theorem, for the Hilbert class field (i.e. maximal unramified Abelian extension) Artin's ...
0
votes
1answer
40 views

Prime decomposition in $\mathbb Z[x]/(x^3-x^2+x+1)$

If $K$ is the unique number field of discriminant $-44$, K is isomorphic to the field generated over $\mathbb Q$ by a root of the polynomial $x^3-x^2+x+1$ with $\mathcal O_K=\mathbb ...
1
vote
0answers
54 views

Show that $p$ is inert in the ring of integers of $K$

Let $p$ be a prime, $n$ a positive integer, $E$ a finite field with $p^n$ elements, and $\alpha ∈ E$ an element satisfying $\mathbb F_p(\alpha)=E$. If $\bar f$ is the minimal polynomial of $\alpha$ ...
0
votes
1answer
38 views

Primitive element theorem for finite fields

Primitive element theorem for finite fields Can you explain $2$ points in the proof of the proposition below $\bullet$ First $\alpha$ is the root of the polynomial $T^{p^s}-T$, because $\mathbb ...
3
votes
3answers
38 views

Split 16 Consecutive Integers into Two Subsets of 8 Integers

Show that any given set of sixteen consecutive integers {$x+1,x+2,\ldots,x+16$} can be divided into two eight element subsets with the properties that they have the same sum, the sums of the squares ...
2
votes
1answer
18 views

Presented matrix and number ring.

Let $V$ be the module generated by the column matrix $A= (2, 1+ \sqrt{-5})^T$. Prove that the residue of $A$ in $\mathbb{Z}[\sqrt{-5}]/ \mathfrak{P}$ has rank 1 for every prime ideal $\mathfrak{P}$ of ...
2
votes
1answer
24 views

Explicit calculation of residue field in Cyclotomic integers

I would like to show that $(1-\zeta)$ is a prime ideal in $\mathbb{Z}[\zeta]$, where $\zeta=\zeta_p =e^{\frac{2\pi i}{p}}$, for a prime $p$. I am aware that we can show $(1-\zeta)=(1-\zeta^i)$, for ...
1
vote
1answer
35 views

Find a counterexample to the following lemma if we change the statement slightly.

let K be an algebraic number field and let $O_K$ be its ring of integers. Lemma; Let $a,b$ be fractional ideals of $O_K$. If $b \subseteq a$ then there is an ideal $c$ such that $b=ac$. I need to ...
4
votes
0answers
76 views

Mordell Equation $y^2 = x^3 - 20$. [closed]

Prove that the only integral solutions to $y^2 = x^3 − 20$ are $(x, y) = (6, \pm14)$.
8
votes
4answers
83 views

Show that $\mathcal{O}_K$ is not UFD with $K = \mathbb{Q}(\sqrt{-13})$

Let $K = \mathbb{Q}(\sqrt{-13})$. Show that its ring of integers $\mathcal{O}_K$ is not an UFD. $-13 \equiv 3 \bmod{4}$, so $\mathcal{O}_K = \mathbb{Z}\bigl[\sqrt{-13}\bigr]$. We will use the ...
1
vote
1answer
33 views

Can someone prove or help me understand the following about Euclidean fields?

Why is it that if $\delta$ and $\delta'$ both divide $\alpha$ and $\beta$, and that every $\gamma$ which divides $\alpha$ and $\beta$ also divides $\delta$ and $\delta'$, then $\delta$ and $\delta'$ ...
6
votes
2answers
98 views

Is $3$ a prime element of $\mathbb{Z[\eta]}$?

How to check whether $3$ is a prime element or not in $\mathbb{Z[\eta]}$, where $\eta$ is a $17$th primitive root of unity. Also in general how can we check an element is prime or not in ...
3
votes
1answer
50 views

Frobenius automorphism and cyclotomic extension

There is a lemma from a lecture I attended where I scribbled down notes and try to make sense of the proof afterwards and there is a spot at which I am stuck. First I have to set up some notations and ...
3
votes
0answers
30 views

Direct Proof of Divisibility in Extensions of Number Fields

Let $L/K/\mathbb Q$ be a tower of number fields. The result I want to show is that the discriminant $\Delta_K$ divides the discriminant $\Delta_L$. I was wondering if there was a "direct" proof of ...
2
votes
1answer
15 views

Discriminant of a product of polynomials

Let $f,g$ be irreducible, monic and in $\mathbb Z[x]$. Then (I hope this is correct) $disc(f\cdot g)=disc(f)\cdot disc(g)\cdot\prod_i\prod_j(a_i-b_j)^2$ where the $a_i$ are the roots of $f$ ...