Questions related to the algebraic structure of algebraic integers

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3answers
250 views

How to show that $1-\zeta$ is prime in the order $\{ 1, \zeta, \ldots, \zeta^{l-2} \}$?

I am trying to prove the following: Let $l$ be a prime and let $\zeta$ be a $l$th root of unity. Show that, in the order $\{ 1, \zeta, \ldots, \zeta^{l-2} \}$ of the field $\mathbb{Q}(\zeta)$, if ...
8
votes
1answer
127 views

Geometric intuition behind the Hasse principle

Let $f(X,Y) \in \mathbb{Q}[X,Y]$ be a quadratic polynomial. The Hasse-Minkowski theorem says that $f(X,Y) = 0$ has a solution $(x,y) \in \mathbb{Q}^2$ iff it has a solution in $\mathbb{R}^2$ and ...
8
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2answers
255 views

Some questions about ramifications of primes

I was trying to show that the ring of integers of $K=\mathbb{Q}(\sqrt[3]{2})$ is $\mathbb{Z}[\sqrt[3]{2}]$ and came up with the following question. Computing the discriminant of ...
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3answers
663 views

Enlightening proof that the algebraic numbers form a field

The proof I'm familiar with that the algebraic numbers $\mathbb A$ form a field uses the fact that the resultant of two polynomials $p,q\in\mathbb Q[x]$ satisfies the following properties: It is $0$ ...
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3answers
860 views

Prime ideals in the ring of algebraic integers

Let $\mathcal{O}$ be the ring of all algebraic integers: elements of $\mathbb{C}$ which occur as zeros of monic polynomials with coefficients in $\mathbb{Z}$. It is known that $\mathcal{O}$ is a ...
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5answers
1k views

Minimal polynomial of $\frac{\sqrt{2}+\sqrt[3]{5}}{\sqrt{3}}$

I am struggling to find the minimal polynomial for $\displaystyle \frac{\sqrt{2}+\sqrt[3]{5}}{\sqrt{3}}$. Does anyone have any suggestions? Thanks, Katie.
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3answers
522 views

Intuition regarding Chevalley-Warning Theorem

Three versions of the theorem are stated on pages 1-2 in these notes by Pete L. Clark: http://math.uga.edu/~pete/4400ChevalleyWarning.pdf Could anyone offer some intuitive way to think about this ...
7
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4answers
353 views

Is there a procedure to determine whether a given number is a root of unity?

Let $z$ be an algebraic number of modulus one. Is there a finite procedure that tells us whether $z$ is a root of unity? EDIT: As TonyK and David asked, what I had in my mind is $z$ such that I have ...
7
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2answers
364 views

Classgroup of $\mathbb{Q}(\sqrt{2},\sqrt{-13})$

How would you compute the classgroup of the biquadratic number field $\mathbb{Q}(\sqrt{2},\sqrt{-13})$? I would prefer a method as "from scratch" as possible. Please avoid, if possible, quoting ...
7
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1answer
333 views

Any resource of the applications of the theory of class fields

We all agree that the theory of class fields plays an eminent role in modern number theory. Nevertheless, what was our main concern is that how to solve various Diophantine equations to which the ...
7
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2answers
583 views

Is the algebraic closure of a $p$-adic field complete

Let $K$ be a finite extension of $\mathbf{Q}_p$, i.e., a $p$-adic field. (Is this standard terminology?) Why is (or why isn't) an algebraic closure $\overline{K}$ complete? Maybe this holds more ...
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5answers
301 views

Proving $\sqrt{2}\in\mathbb{Q_7}$?

Why does Hensel's lemma imply that $\sqrt{2}\in\mathbb{Q_7}$? I understand Hensel's lemma, namely: Let $f(x)$ be a polynomial with integer coefficients, and let $m$, $k$ be positive integers ...
7
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2answers
178 views

Why study integrality?

Here are a few of the basic definitions related to integrality. (1) A polynomial in $R[x]$ is monic if its leading coefficient is $1$. (2) An element is integral over a ring $R$ if it ...
7
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2answers
168 views

Show field of fractions is finite extension of $\mathbb{Q}$

Let $A$ be a ring which is also a finitely generated $\mathbb{Z}$-module. If $A$ is an integral domain and $K$ is its field of fractions and $K$ has characteristic zero, then why is $K$ a finite ...
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3answers
583 views

Preparations for reading Algebraic Number Theory by Serge Lang

I am eager to learn algebraic number theory. It seems that Serge Lang's Algebraic Number Theory is one of the standard introductory texts (correct me if this is an inaccurate assessment). I flipped ...
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2answers
468 views

Class field theory for function fields and a curious statement

Let $X_0$ be a smooth curve over a finite field $\mathbb{F}_q$, and let $X$ be the base-change to the algebraic closure. I read that, according to class field theory in function fields, "the image ...
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3answers
206 views

Integers of the form $x^2-ny^2$

Is there any algorithm to represent the given integers in the form $N=x^2-ny^2$, $n>1$, say for $n=2$. As we have for any prime, $p$, a prime $q$ can be written as $q=x^2+py^2$,whenever $q$ is ...
7
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2answers
207 views

Solve: $x^2-py^2=q$

Solve $$x^2-py^2=q$$ for integers $x,y$, here $p,q$ are both given prime numbers. It's obvious that $p,q$ should satisfy $(\frac{p}{q})=(\frac{q}{p})=1,$ here $(\frac{p}{q})$ is the Jacobi symbol. ...
7
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1answer
133 views

Are there infinitely many pairs of rational numbers such that…

Are there infinitely many pairs of rational numbers $(a,b)$ such that $a^3+1$ is not a square in $\mathbf{Q}$, $b^3+2$ is not a square in $\mathbf{Q}$ and $b^3+2 = x^2(a^3+1)$ for some $x$ in ...
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2answers
332 views

Approximation Lemma in Serre's Local Fields

Let $A$ be a Dedekind domain, and let $K$ be its field of fractions. In Serre's Local Fields, the following Lemma is stated. Approximation Lemma Let $k$ be a positive integer. For every $i$, ...
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2answers
433 views

Elementary proof of $\mathbb{Q}(\zeta_n)\cap \mathbb{Q}(\zeta_m)=\mathbb{Q}$ when $\gcd(n,m)=1$.

In an answer to another question I used the fact that $\mathbb{Q}(\zeta_m)\subseteq \mathbb{Q}(\zeta_n)$ if and only if $m$ divides $n$ (here $\zeta_n$ stands for a primitive $n$th root of unity, ...
7
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2answers
323 views

Roots of unity in $\mathbb{Q} _{11}$

Here $\mathbb{Q} _{11}$ denotes the 11-adic field. How can I show that the only root of unity of order 7 in this field is 1? Is it true that for any two distinct primes $p,q$, the only root of unity ...
7
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1answer
100 views

What do we know about the class group of cyclotomic fields over $\mathbb{Q}$?

Motivated by this question, I am curious how one can characterize primes that splits completely in the Hilbert class field of $\mathbb{Q}(\zeta_q)$, where $q$ is a prime. Then I realize how much I ...
7
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1answer
115 views

Infinite $p$-extension contains $\mathbb{Z}_p$-extension

Does the Galois group of every infinite $p$-extension $K$ of a number field $k$ contain a (closed) subgroup such that the quotient group is isomorphic to $\mathbb{Z}_p$? My feeling is "yes", but I'm ...
7
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1answer
193 views

Residue at $s=1$ for $\zeta$-functions

Is there any sort of a bound for the magnitude of this residue? I've been looking at some algebraic number theory problems from Princeton's general exam. One of them is the following: Why does a ...
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1answer
344 views

why does a certain formula in Lang's book on modular forms hold?

Background: Let $k$ be an even integer. The Eisenstein series are defined by $$E_{k} = 1 - \frac{2k}{B_{k}}\sum_{n=1}^{\infty} \sigma_{k-1}(n)q^{n}$$ where $$\sigma_{k-1}(n)= \sum\limits_{d \mid ...
7
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2answers
523 views

A question on FLT and Taniyama Shimura

Sometime back i watched the documentary of Andrew Wiles proving the Fermat's Last theorem. A truly inspiring video and i still watch it whenever i am in a depressed mood. There are certain ...
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2answers
659 views

Ideal class group of $\mathbb{Q}(\sqrt{-103})$

I want to calculate ideal class group of $\mathbb{Q}(\sqrt{-103})$. By Minkowsky bound every class has an ideal $I$ such that $N(I) \leq 6$. It is enough to consider prime ideals with the same ...
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1answer
870 views

What is the discriminant of a quadratic extension over a number field?

Let $K$ be a number field and $d \in \mathcal{O}_K \setminus \mathcal{O}_K^2$. What is the discriminant of the extension $K[\sqrt{d}]/K$ ? Do we know its ring of integers and which primes are split or ...
7
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1answer
628 views

Reference book for Artin-Schreier Theory

The aim of the question is very simple, I would like to study Artin-Schreier Theory, but I have had embarassing difficulties in finding a book which could help me in doing that. In specific I'm ...
7
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1answer
227 views

Computing Brauer characters of a finite group

I am studying character theory from the book "Character Theory of Finite Groups" by Martin Isaac. (I am not too familiar with valuations and algebraic number theory.) In the last chapter on modular ...
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2answers
93 views

Primes inert in quadratic field of class number one

Let $K = \mathbb{Q}(\sqrt{-d})$ be an imaginary quadratic field of class number one (i.e. every ideal in $\mathcal{O}_K$ is principal, i.e. $\mathcal{O}_K$ is a principal ideal domain). Let $d_K$ be ...
7
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1answer
220 views

All number fields with absolute value of discriminant $\le 20$

I need to find all number fields with absolute value of discriminant $\le 20$. Using Minkovsky's theorem I understood that it should be quadric or cubic extension. The case of quadric is very ...
7
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1answer
222 views

Computing the ring of integers of a number field

This question arose from the need to see the splitting behaviour of primes over an extension of number fields. One criterion is the Kummer's theorem, which gives the decomposition of the base prime in ...
7
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1answer
196 views

Principal ideal domain not euclidean

Can anyone give an example of a principal ideal domain that is not Euclidean and is not isomorphic to $\mathbb{Z}[\frac{1+\sqrt{-a}}{2}]$, $a = 19,43,67,163$? I believe it is conjectured that no ...
7
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1answer
104 views

Proof in Kummer Theory - why is this subgroup finite?

I'm doing a course on elliptic curves. We're working with a field $K$ with $\mu_n \subset K$ ($K$ contains all $n$th roots of unity) and $\mbox{char}(K)\not|\;n$. I'm trying to understand the proof ...
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1answer
162 views

Determining when a certain binomial sum vanishes

Consider the following sum of signed binomial coefficients: $$S_{n,a,p} = \sum_{i \equiv a \mod p} \binom{n}{i}(-1)^i$$ ($n$ is a positive integer, $p$ is an odd prime, $a$ is between $0$ and ...
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1answer
175 views

How can I determine in practice whether two elliptic curves over $\mathbb{Q}$ have isomorphic $p$-torsion?

Let $E_1$ and $E_2$ be elliptic curves over $\mathbb{Q}$ with good, ordinary reduction at an odd prime $p$. I'm wondering how to determine whether $E_1[p]$ and $E_2[p]$ are isomorphic ...
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1answer
346 views

Special privilege enjoyed by Elliptic Curves with Complex Multiplication

I think after reading the title one may understand the intention of me, this question is concerned about the Elliptic curves having a Complex Multiplication. I have been reading many theorems, ( ...
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2answers
560 views

About cyclotomic extensions of $p$-adic fields

I've been working on the problem of finding the maximal abelian extension of $\mathbb{Q}_5$ that is killed by $5$. In other words, find the abelian extension of $\mathbb{Q}_5$ with Galois group ...
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1answer
106 views

Implications between $(\frak{a} \cap \frak{b})$ = $\frak{a} \frak{b}$ and $(\frak{a})$ + $(\frak{b})$= $(1)$

In a general commutative ring, $(\frak{a} \cap \frak{b})$ = $\frak{a} \frak{b}$ does not imply ($\frak{a}$) + ($\frak{b}$) = ($1$); whereas ($\frak{a}$) + ($\frak{b}$) = ($1$) does imply $(\frak{a} ...
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1answer
218 views

Doesn't Abhyankar's lemma contradict faithful flat descent (no, but I'm confused)

Let $O_K$ be a dvr with fraction field $K$. Let $L/K$ be a tamely ramified finite Galois extension. Then, Abhyankar's Lemma implies that there exists a finite Galois extension $K^\prime/K$ such that ...
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1answer
117 views

“Real part” of a number field

Let ${\mathbb K} \subseteq {\mathbb C}$ be a finite extension of $\mathbb Q$, and let $n=[{\mathbb K} : {\mathbb Q}]$. Let $X_{\mathbb K}$ denote the set of all “components” (i.e., real and imaginary ...
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1answer
183 views

The number of solutions of the equation $x^n+y^n=1$ over $\mathbb{Z}_p$ with $\mathbb{Z}_p=\{\alpha\in \mathbb{Q}_p:|\alpha|_p\le 1\}$.

The number of solutions of the equation $x^n+y^n=1$ over $\mathbb{Z}_p$ with $\mathbb{Z}_p=\{\alpha\in \mathbb{Q}_p:|\alpha|_p\le 1\}=\{\sum_{i=0}^\infty a_ip^i:0\le a_i \le p-1\}$. I was trying ...
7
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1answer
218 views

How do we know if there are any better bounds than the Minkowski bound?

This question may be an exact replicate of some earlier question elsewhere. I want to know if there are better bounds than the Minkowski bound for the norm of the smallest ideal in an ideal class of ...
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1answer
217 views

Degrees of the real and imaginary parts of an algebraic number

I am working on a theory of generalized geometric constructions, which involves generating new numbers as real roots of polynomials whose coefficients are existing numbers satisfying certain ...
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1answer
197 views

Show the two fields are not isomorphic

Let $p,q,r$ be prime integers with $q\neq r$. Let $\sqrt[p]q$ denote any root of $x^p-q$ and $\sqrt[p]r$ denote any root of $x^p-r$. Please show that $\mathbb{Q}(\sqrt[p]q)\neq\mathbb{Q}(\sqrt[p]r)$. ...
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1answer
270 views

How does the Artin symbol generalize Legendre and Hilbert symbols?

I am currently going through a course in Class Field theory and have read that the Artin symbol generalizes the Legendre and Hilbert symbols. I was wondering what field extensions to consider to see ...
7
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1answer
108 views

Characterizing a sequence of primes

This is an attempt to finish up Characterizing the primes which don't divide any Pell-Lucas number(s) For primes $p \equiv 3 \pmod 4,$ there is always some solution to $x^2 - 2 y^2 = \pm 1$ with ...
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1answer
93 views

Volume of first cohomology of arithmetic complex

Let $K$ be a number field and consider the Arithmentic complex $\Gamma_{Ar}(1)^\bullet$ be defined by $$\begin{array} A\Bbb R^{r_1+r_2} & \stackrel{\Sigma}{\longrightarrow} & \Bbb R \\ ...