Questions related to the algebraic structure of algebraic integers

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1answer
35 views

How to construct a polynomial from a radix-term?

A term only composed of the following operatings shall henceforth be called a radix term because I don't know how these terms are called. A radix term $t$ is either an integer or a sum of two radix ...
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24 views

Lattice with conditions

In Book's Algebraic number theory, I. Stewart page 142: Theorem 7.2: If $p$ is prime of the form 4k+1 then $p$ is sum of two squares. Proof: The multiplicative group $G$ of the field $\mathbb ...
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1answer
35 views

Field, algebraic element

1) Let $E/F$ an extension and let $\alpha,\beta\in E$ be algebraic elements over $F$. If $\alpha\neq 0$, prove that $\alpha+\beta$, $\alpha\beta$ and $\alpha^{-1}$ are all algebraic over $F$. 2) If ...
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2answers
72 views

Example of non fractional ideal

Let $R$ be an integral domain with fraction field $K$, and let $I$ be an $R$-submodule of $K$. We say that $I$ is a fractional ideal of $R$ if $rI\subset R$ for some nonzero $r \in R$. My question ...
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1answer
48 views

Is an extension of a discrete absolute value discrete too?

Suppose $L/K$ is a finite extension of fields, suppose $v$ is a non-archimedean absolute value on $L$ such that the restriction of $v$ on $K$ is non-trivial and discrete. Can we say that $v$ is ...
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1answer
21 views

Gaussian sums values

I have the following problem: Denoting $S(q,a,\chi ) = \sum_{x=1}^q \chi (x) e(ax/q)$, where $\chi $ is an arbitrary character modulo $q$, I have to prove $$\sum_{a=1}^q \vert S(q,a,\chi ) \vert ^2 = ...
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2answers
55 views

Prove all elements of $A$ is algebraic over $C$, if all elements of $A$ are algebraic over $B$ and $B$ are algebraic over $C$

Let there be 3 fields $A$, $B$ and $C$. If all elements of $A$ are algebraic over $B$ and all elements of $B$ are algebraic over $C$, prove that this implies that all elements of $A$ is algebraic ...
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1answer
69 views

Sum of roots of unity an algebraic integer proof

Let S be the sum of a finite number of nth roots of unity (where n is fixed, and the sum is non-zero). How do I go about showing that S is an algebraic integer in the cyclotomic field of order n ?
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45 views

Polynomial Diophantine Equations

So in general how does one decide if: $$ a_0 + a_1x + a_2x^2 ... a_{n_1}x^{n_1} = b_1y + b_2y^2 ... + b_{n_2}y^{n_k}$$ Has solutions for integers $x,y$ given real numbers $a_0, a_1. .. a_{n_1}, b_1 , ...
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67 views

Show that this is a finitely generated abelian group and compute its rank

If $n$ is a square-free integer such that $n >1$, and $ K = \mathbb Q ( \sqrt n )$. Let $ A_K$ the ring of algebraic integers. Show that $ A_K ^ \times $ is a finitely generated abelian group and ...
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37 views

Prove that $D(\alpha)=D(\beta)$

Let K be an algebraic number field. Let $\alpha \in$ K. Let $\beta$ be conjugate of $\alpha$ relative to K . Prove that $D(\alpha)=D(\beta)$. $D(\alpha)$:= Let K be algebraic number field of degree ...
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1answer
18 views

Norm of $(\alpha - a) = (-1)^{\deg f}f(a)$

Let $\Bbb Q(\alpha)$ be a number field, and $f$ the minimal polynomial of $\alpha$. Why is $N_{\Bbb Q(\alpha)/\Bbb Q} (\alpha-a)= (-1)^{\deg f}f(a)$? This works obviously for $a=0$ by the definition ...
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1answer
99 views

$(p)$ is a prime ideal in $\mathbb{Z}[\sqrt{n}]$ if and only if $n$ is not a square mod $p$

Let $n$ be a square-free integer such that $n\equiv 2,$ or $3\bmod 4$. If $p\in\mathbb{Z}$ be a prime such that $n$ is not a square modulo $p$, then $(p)$ is a prime ideal in ...
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1answer
52 views

Questions on a proof of “All prime ideals of a Dedekind domain are invertible”

I tried to prove this theorem : All prime ideals of a Dedekind domain is invertible. i.e, For every prime ideal $\mathfrak{p}$ of Dedekind domain $R$, there exists $\mathfrak{p}^{-1} \subseteq ...
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2answers
45 views

number cubic polynomials possible

Let $p(x)$ be a cubic polynomial with integral coefficients , such that $p(a)=b$, $p(b)=c$, $p(c)=a$ for $a,b,c$ being distinct integers . find number of such possible polynomials.
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113 views

Residue class field of ring of integers is finite

Let $\mathcal{o}_K$ be a ring of algebraic integers. I have a proof for the fact that $\mathcal{o}_K$ is a free module of finite rank over $\mathbb{Z}$. Now, let $\mathcal{p}$ be a prime ideal of ...
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1answer
60 views

Prove that $A^B=\Bbb{Z}(\sqrt{2})$

$A^B =\{\,b \in B : \text{$b$ is integral over $A$}\,\}$ Let $A=\Bbb{Z}$ and $B=\Bbb{Q}[\sqrt{2}]$ . Prove that $A^B=\Bbb{Z}[\sqrt{2}]$ Thank you for your helping. i am sorry for my poor English ...
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54 views

Infinite primes and notation

While reading a book about algebraic number theory, the symbol for a rational prime $p$ $$p^\infty$$ often occurs and I was wondering, what the exact definition of this is. Also, what is the ...
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1answer
45 views

$p$ is an odd prime of the form $p=x^2+2y^2$ iff $p\equiv_8$ $1$ or $3$ [duplicate]

How would I prove the following: Show that an odd prime $p$ can be written on the form $p=x^2+2y^2$ for some $x,y\in\mathbb Z$ iff $p\equiv_8 1, 3$. Hint: use the quadratic reciprocity and the ...
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1answer
18 views

The basis for additive subgroup is discriminant-invariant

i.e. given bases $\{\beta_{i}\}$ and $\{\gamma_{i}\}$ for S additive subgroup of number field K (degree n over $\mathbb{Q}$), then $disc(\{\beta_{i}\})=disc(\{\gamma_{i}\})$. any hints? Is that ...
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1answer
65 views

How to show that $|z|=(z\overline{z})^{1/2}$ is the only valuation of $\mathbb{C}$ that extends the absolute value of $\mathbb{R}$?

How can I show that $|z|=(z\overline{z})^{1/2}$ is the only valuation of $\mathbb{C}$ that extends the absolute value of $\mathbb{R}$? I can see that it suffices to work with the unit disk, i.e. it ...
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1answer
40 views

if $\beta\in \mathcal{O}\cap\mathbb{Q}[\mathbb{w}]$, then $\beta=a_{0}+a_1w+…+a_{p-1}w^{p-1}$

$w=e^\frac{2i}{p}$ where p is odd prime. $\mathbb{Z}[]$ if $\beta\in \mathcal{O}\cap\mathbb{Q}[\mathbb{w}]$, then $\beta=a_{0},+a_1w+...+a_{p-1}w^{p-1}$ where $a_i's$ are unique integers this a ...
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1answer
100 views

Power series ring over a ring of integers

Let $K/\mathbb {Q}_p$ be a finite extension, $\mathcal{O} := \mathcal{O}_K$ the ring of integers of $K,$ $\frak p$ the maximal ideal of $\mathcal{O}$, and $\pi$ a uniformizer, i.e., $\frak{p} = ...
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1answer
40 views

Gauß sum and primitive character

I am working with Daniel Marcus "Number Field" Book. And I have a question to the following Lemma: $$\tau_k(\chi)=\left\{\begin{array}{ll} \bar\chi(k)\tau(\chi), & \textrm{if }(k,m)=1 \\ ...
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1answer
28 views

$L/F$ unramified , $\sigma(L)/\sigma(F)$ it is also unramified

Let $L/K$ be an extension of field, and $F$ be a subfield of $L$ contains $K$ such that the extension $L/F$ is an unramified Galois extension, if $\sigma$ is an isomorphism of $L$ leaving $K$ fixed, ...
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1answer
196 views

How to calculate such sums of Legendre symbols?

How to calculate such sums as $\sum_{x\in\mathbb{F}_p} \left(\frac{x^2+ax+b}{p} \right)$ If $x^2+ax+b$has a root, $b$ may be eliminated and the sum is evaluated to be $0+\sum_{x\in\mathbb{F}_p^*} ...
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1answer
314 views

SageMath: Embed all roots of a polynomial

I have a polynomial, and I would like to make a number field in Sage that would contain all roots of this polynomial. I have tried the following code but it throws an error: ...
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1answer
105 views

Points on elliptic curves

I am learning elliptic curves theorem and I have read in more papers that for two distinct points $P$ and $Q$ there is always point $R$ such that $P+Q+R = \infty$. I know that this point should be ...
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1answer
51 views

Example of number field $K$ such that $[K(\zeta_m):K] < \phi(m)$

Actually two questions: 1) What is an example of number field $K$ such that $[K(\zeta_m):K] < \phi(m)$? 2) In class we discussed that if $L$ is a field of characteristic zero, and $K$ is the ...
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1answer
179 views

On the norm formula $N(IJ) = N(I)N(J)$ in an order of an algebraic number field

Let $K$ be an algebraic number field of degree $n$. Let $\mathcal{O}_K$ be the ring of algebraic integers of $K$. Let $R$ be an order of $K$, i.e. a subring of $K$ which is a free $\mathbb{Z}$-module ...
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1answer
45 views

Explain this Modular Arithmetic Expression in Z[i]

Let $\pi = a+bi$ and $\lambda = c+ di$ be relatively prime in $\mathbb{Z}[i]$. They also said that they were "primary" meaning that $\pi = \lambda = 1 (\text{mod } (1+i)^3)$, though I suspect this is ...
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1answer
45 views

Two discrete valuations on a Dedekind domain

A question from Frohlich and Taylor's book 'Algebraic Number Theory', page 60. Let $\mathfrak o$ be a Dedekind domain with quotient field $K$ and let $v$ be a discrete valuation on $K$. Let ...
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1answer
78 views

Quadratic fields with cyclic class group

Let $\mathbb{K}$ be a real quadratic field, with discriminant $d_{\mathbb{K}}<36$. Then the Minkowski's bound is $\frac{1}{2}\sqrt{d_{\mathbb{k}}}<3$. By the Minkowski's Theorem, each ideal ...
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2answers
67 views

When $a$ is even, the difference between $(a/2) \mod N$ and $(a \mod N)/2$?

folks. Could I ask for your help? Let $N$ be a positive integer and $a$ be an even integer, i.e., $a=2x$ for an integer $x$. Then think of $W_N^{\frac{a}{2}}$, where $W_N=e^{j\frac{2\pi}{N}}$. ...
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1answer
58 views

Product of all ideals of prime norm

I am unsure about the truth of the answer in How to find all the ideals of a given norm?, where it is claimed "The ideal (p) is the product of all of the prime ideals of norm a power of p (with ...
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1answer
139 views

Show that $B$ is a basis for $K$ as a vector space over $\mathbb{Q}$

Let $K$ be a number field of degree $d$, and let $B = \{b_1,\ldots,b_d\}$ be a subset of $K$ of cardinality $d$ such that the matrix $(\mathrm{Tr}(b_ib_j))^d_{i,j=1}$ has non-zero determinant ...
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1answer
42 views

If ideals $Q_1,Q_2$ lie over a prime in $\Bbb{Z}$ their product lies over the prime squared?

Suppose we have a Dedekind domain $R$ which for the moment we can take to be $\mathcal{O}_K$ for some algebraic number field $K$. Now suppose that $Q_1,Q_2$ are prime ideals that lie over a prime ...
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2answers
64 views

Lattice of max. rank - compact in quotient topology - bounded subset

$L\subset \mathbb{R}^{n}$ lattice of max. rank $\Leftrightarrow \mathbb{R}^{n}/L$ compact in quotient topology $\Leftrightarrow \exists$ bounded subset $B\subset\mathbb{R}^{n}$ s.t. $L+B = ...
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1answer
59 views

Why does $\mathfrak{o}_L \cong \mathfrak{o}_K^n$ imply that $\mathfrak{o}_L/\pi_k\mathfrak{o}_L$ is a $k_K$ vector space of dimension $n$?

I'm trying to read a proof of the following proposition: Let $L/K$ be a finite, separable extension of a complete discretely valued field. Then $e(L/K)f(L/K)=[L:K]$. I'm stuck on the step: ...
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1answer
157 views

Proving that a Dirichlet character $\chi$ must be the Kronecker symbol $\chi_D = \left ( \frac{D}{\cdot} \right )$

I'm stuck trying to prove that a particular Dirichlet Character $\chi$ must be the Kronecker symbol $\chi_D = \left ( \frac{D}{\cdot}\right )$. The context is the following. Let $d$ be a square-free ...
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2answers
152 views

Ideal quotient of fractional ideals

Wat exactly does the notation $$A : (BC)$$ mean? We are talking about fractional R ideals. From definitions i get, if $$a \in A:(BC)$$ then $$a(BC)\subset A$$ But what am i saying exactly? I hope ...
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2answers
163 views

The discriminant of an integral binary quadratic form and the discriminant of a quadratic number field

Let $ax^2 + bxy + cy^2$ be a binary quadratic form over $\mathbb{Z}$. Let $D = b^2 - 4ac$ be its discriminant. It is easy to see that $D \equiv 0$ (mod $4$) or $D \equiv 1$ (mod $4$). Conversely ...
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1answer
600 views

Uniqueness, units of the Eisenstein Integers

Let $\zeta$ be the cube root of 1 given by $\zeta=\frac{-1}{2}+i\frac{\sqrt{3}}{2}$ and let $\mathbb{Z}[\zeta]=\{a+\zeta b: a, b\in \mathbb{Z}\}$, called the "Eisenstein integers". How prove the ...
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1answer
107 views

Generalizations of Monogenic Fields

We recall that a number field $K$ is called monogenic if there exists some $\alpha \in \mathcal{O}_K$ such that $\mathcal{O}_K = \mathbb{Z}[\alpha]$. If instead we take a tower of finite extensions ...
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1answer
210 views

Integral ideals of norm less than the Minkowski Bound

Consider a number field $K$ and suppose I want to find the class group and class number of $K$. One of the first steps is to compute the the Minkowski bound. Suppose our bound is $B$. In all the ...
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1answer
9 views

$K_S$ modulo $K_S^n$, where $K_S$ is the group of $S$-units

Let $K$ be a field containing the $n$th roots of unity, $S$ a finite set of places containing all the archimedean ones, and $K_S$ the group of $S$-units, i.e. those $x \in K^{\ast}$ which are units at ...
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1answer
22 views

the field fixed by inertia group is the maximal unramified field

$K/F$ is a ablian galois extension of a number field. $\mathcal{O}_F$ (resp.$\mathcal{O}_K$) is the interger ring of $F$(resp.$K$). Let $\mathfrak{P}$ be a prime of $\mathcal{O}_K$ and ...
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0answers
21 views

square classes of quadratic extensions of 2-adic fialds

I have a question about square classes of quadratic extensions of 2-adic fields. I appreciate anybody help me to understand. Why all elements of $1+\mathfrak{p}^5$ are square in ...
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0answers
26 views

Discriminant ideal and short exact sequence of finite group schemes

Let $0 \to G' \to G \to G'' \to 0$ be a short exact sequence of finite flat commutative group schemes over a Dedekind domain $\mathcal O$ with field of fractions of characteristic zero. Let ...
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50 views

Non maximal prime ideals and localization

In the chapter on localization in Neukirch, Algebraic Number Theory, the following is mentioned: (A is an integral domain) Usually $S$ will be the complement of a union $\bigcup_{\mathfrak p \in ...