Questions related to the algebraic structure of algebraic integers

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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 ...
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879 views

How can one express $\sqrt{2+\sqrt{2}}$ without using the square root of a square root?

I was trying to review some analysis, and came across problem 3 from page 78 of Walter Rudin's Principles of Mathematical Analysis. As part of the problem, I wanted to try to write ...
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3answers
422 views

Attaining the norm of an ideal in a number field by the norm of an element

Let $K$ be a number field of degree $n$ and $\mathfrak{a}$ be an ideal in its ring of integers $\mathcal{O}_K$. We can consider: The norm $N(\mathfrak{a})$ of $\mathfrak{a}$. The norms $N(x)$ of the ...
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483 views

Unique factorization in $\mathbb Z(\sqrt{-19})$

An elementary confusion about class number: In $\mathbb Z(\sqrt{-19})$ we have $N(1+\sqrt{-19}) = (1+\sqrt{-19})(1-\sqrt{-19}) = 2^2\cdot 5.$ I see that 2 and 5 are irreducible, 4 is not. In a ...
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379 views

A simple question about Iwasawa Theory

There has been a lot of talk over the decades about Iwasawa Theory being a major player in number theory, and one of the most important object in said theory is the so-called Iwasawa polynomial. I ...
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204 views

Elements of cyclotomic fields whose powers are rational

Suppose the polynomial $t^k - a$ has a root (hence splits) in $\mathbb{Q}(\zeta_k)$. For which $k$ does it follow that one of the roots of $t^k - a$ is rational? In particular, are there infinitely ...
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236 views

Can the argument of an algebraic number be an irrational number times pi?

This is mainly out of curiosity. Let $\nu$ be an algebraic number. Can Arg($\nu$) be of the form $\pi \times \mu$ for an irrational number $\mu$?
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253 views

Is the ring of p-adic integers of finite type over the ring of integers?

Denote by $\mathbb{Z}_p$ the ring of $p$-adic integers. Is $\mathrm{Spec}(\mathbb{Z}_p)$ of finite type over $\mathrm{Spec}(\mathbb{Z})$?
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Good undergraduate level book on Cyclotomic fields

I have Lang's 2 volume set on "Cyclotomic fields", and Washington's "Introduction to Cyclotomic Fields", but I feel I need something more elementary. Maybe I need to read some more on algebraic number ...
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587 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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How to find the integral closure of $\mathbb{Z}_{(3)}$ in the field $\mathbb{Q}(\sqrt{-5})$?

Let $v$ be the 3-adic valuation on $\mathbb{Q}$ and consider the subring $\mathbb{Z}_{(3)}$ of $\mathbb{Q}$ defined by $$ \mathbb{Z}_{(3)} = \{ x \in \mathbb{Q} : v(x) \geq 0 \}. $$ That is, ...
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Unique quadratic subfield of $\mathbb{Q}(\zeta_p)$ is $\mathbb{Q}(\sqrt{p})$ if $p \equiv 1$ $(4)$, and $\mathbb{Q}(\sqrt{-p})$ if $p \equiv 3$ $(4)$

I want to prove the assertion: The unique quadratic subfield of $\mathbb{Q}(\zeta_p)$ is $\mathbb{Q}(\sqrt{p})$ when $p \equiv 1 \pmod{4}$, respectively $\mathbb{Q}(\sqrt{-p})$ when $p \equiv 3 ...
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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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1answer
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Is $\mathbb Z[\sqrt{-3}]$ Euclidean under some other norm?

I know that $\mathbb Z[\sqrt{-3}]$ is not a Euclidean domain under the usual norm $N(x + y\sqrt{-3}) = x^2 + 3y^2$, but that does not necessarily mean that it can't be a Euclidean domain. Is it ...
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1answer
416 views

How does topology enter Number theory and how we can grasp its essence?

In infinite Galois theory,main theorem failed and we get a "Krull topology" to mend the main theorem, we even generalized that to make definition for profinite group. In local class field theory, the ...
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1answer
206 views

How does (21) factor into prime ideals in the ring $\mathbb{Z}[\sqrt{-5}]$?

The text of the exercise is the following: Show that $\mathbb{Z}[\sqrt{-5}]$ is a Dedekind domain, and that the identities $21 = (4+\sqrt{−5}) \cdot (4 − \sqrt{−5})$ and $21 = 3 · 7$ represent two ...
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Proving Fermat's Last Theorem for n=3 using Euler's and Lamé's approach?

Euler and Lamé are said to have proven FLT for $n=3$ that is, they are believed to have shown that $x^3 + y^3 = z^3$ has no nonzero integer solutions. According to Kleiner they approached this by ...
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321 views

How to calculate the local factor at the infinite place of a function field?

First of all, my apologies for the long-winded nature of this question! Yesterday, Mr. Barquero asked an excellent question regarding function fields and number theory: Why is it "easier" ...
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1answer
386 views

Cyclotomic extensions of $\Bbb Q$

Let $n>4$, and $(h,n) = 1$. How to show that $[\mathbb{Q}(\tan 2 \pi h/n):\mathbb{Q}]= \phi(n)$ or $\phi(n)/2$ or $\phi(n)/4$ respectively if $\gcd(n,8)<4$ or $\gcd(n,8)=4$ or ...
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How would you explain a quadratic field to a beginner?

How would you explain a quadratic field to a beginner? Eg. how did the subject first start? All the modern stuff they use to explain it makes it really confusing how one should think about it in more ...
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1answer
816 views

Vague definitions of ramified, split and inert in a quadratic field

Our lecturer defined the following: Let $K=\mathbb Q(\sqrt d)$ be a quadratic field and $p$ a prime number, then (1) $\ p$ is ramified in $K$ if $\mathcal O_K⁄(p)\cong \mathbb F_p [x]⁄(x^2)$ ...
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How many solutions of the equation $x^2-y^2=1$ are there with $x,y\in\mathbb F_{p^n}$?

Let $p>2$ be an odd prime. Let $\mathbb F_{p^n}$ be the field with $p^n$ elements. How many solutions of the equation $x^2-y^2=1$ are there with $x,y\in\mathbb F_{p^n}$? My work: Char $F=p$. ...
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Show that $x^2 + x + 12 = 3y^5$ has no integer solutions.

Show that $x^2 + x + 12 = 3y^5$ has no integer solutions. Use the fact that the class group of $K$ is cyclic of order 5, where $K=\mathbb{Q}[\alpha]$ and $\alpha$ is the root of $x^2-x+12$. We get ...
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1answer
185 views

Quadratic extensions of $\mathbb Q$

This is a question from Lang's ANT, Thm 6, ch.IV, $\S2$. It states that every quadratic extension of $\mathbb{Q}$ is contained in a cyclotomic extension and that it's a direct consequence of the ...
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1answer
383 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 ...
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83 views

Number of Primes in Ring of Integers of a Number Field

In the ring of integers of an algebraic number field, we say that an algebraic integer $p$ is prime if, whenever $p$ divides product of two algebraic integers, $p$ divides one of them. An algebraic ...
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1answer
192 views

Kronecker-Weber Theorem and Finite Fields

Today it occurred to me that every algebraic extension of $\mathbb F_q$ is cyclotomic, as $\mathbb F_{q^n}$ can be gotten by adjoining a $(q^n-1)^{st}$ root of unity. Also, every algebraic extension ...
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1answer
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What's special about characteristic 2?

I'm trying to get the big picture of how bilinear forms and quadratic forms relate over fields $F$ with $char(F) = 2$ and fields with $char(F) \neq 2$. What I gather so far is that if $char(F) \neq ...
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119 views

Isomorphism of Galois groups induces an isomorphism on decomposition groups?

This is a follow up question to this. Say we have Galois extensions $L ,M$ of $\Bbb{Q}$ with $L \cap M = \Bbb{Q}$ and consider the composite $K = LM$. I want to prove that $$\begin{array}{cccccc} f : ...
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1answer
243 views

An exercise involving characters

Suppose $p$ is a prime, $\chi$ and $\lambda$ are characters on $\mathbb{F}_p$. How can I show that $\sum_{t\in\mathbb{F}_p}\chi(1-t^m)=\sum_{\lambda}J(\chi,\lambda)$ where $\lambda$ varies over all ...
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338 views

Counting the Number of Integral Solutions to $x^2+dy^2 = n$

It is a well known result that the number of integer solutions $(x,y), x>0, y\ge 0$ to $x^2+y^2 = n$ is $\sum_{d|n}\chi(d)$, where $\chi$ is the nontrivial Dirichlet character modulo $4$ such that ...
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129 views

Reference for relation between class number of $\Bbb Q[\sqrt{-p}]$ and partial quotients of $\sqrt p$

So in Ireland and Rosen's, "Classical Introduction to Modern Number Theory", they mention the following incredible fact at the end of Chapter 13, section 1. Suppose $p \neq 3$ and $p \equiv 3 \pmod 4$ ...
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1answer
145 views

$\mathbb Z_p$-rank $\leq r_1+r_2-1$ in Leopoldt's conjecture

I am trying to understand Leopoldt's conjecture as formulated in section 5.5 of Washington's "Introduction to Cyclotomic Fields". For an algebraic number field $K$ let $E$ denote the global units, ...
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737 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
179 views

Where is the calculation hiding in this proof about how $p$ splits in $\mathbb{Q}(\zeta_n)$?

I just worked through a proof in Daniel Marcus' book Number Fields that if $p\nmid n$, the inertial degree of any prime ideal of $\mathbb{Q}(\zeta_n)$ lying over $p$ is equal to the order of $p$ in ...
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560 views

Bijection between an ideal class group and a set of classes of binary quadratic forms.

Let $F = ax^2 + bxy + cy^2$ be a binary quadratic form over $\mathbb{Z}$. We say $D = b^2 - 4ac$ is the discriminant of $F$. If $D$ is not a square integer and gcd($a, b, c) = 1$, we say $ax^2 + bxy + ...
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1answer
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Class number of $\mathbb{Q}(\zeta_{11})$

I want to compute the class number of $K=\mathbb{Q}(\zeta_{11})$. The Minkowski bound here is < 59, and looking at the factorisation of primes, we can show that the ideal class group is actually ...
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3answers
277 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 ...
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1answer
151 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 ...
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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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Irreducibility of cyclotomic polynomials via schemes

A few months ago, someone told me there existed a scheme theoretic proof of the irreducibility of cyclotomic polynomials. I've tried coming up with a proof, and when that didn't really yield anything ...
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Numbers represented by a cubic form

EDIT, April 11, 2013: See answer at http://mathoverflow.net/questions/127160/numbers-integrally-represented-by-a-ternary-cubic-form/127295#127295 This is part 2 ( of 25 discriminants of class number ...
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Is the splitting field unramified over a prime $\mathfrak{p}$ if the discriminant of the polynomial is coprime to $\mathfrak{p}$?

Is the following statement true? Let $R$ be a discrete valuation ring with quotient field $K$ and valuation $\nu$. Suppose that $f(x)\in K[x]$ is an irreducible separable polynomial with ...
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6answers
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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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4answers
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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 ...
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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 ...
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327 views

$-1$ as the sum of three squares of algebraic integers in $\mathcal{O}_{\mathbb{Q}(\sqrt{-7})}$

Following the result of Ivan Niven, in his paper http://www.ams.org/journals/tran/1940-048-03/S0002-9947-1940-0003000-5/S0002-9947-1940-0003000-5.pdf That is whenever $d\equiv 3 \pmod 4$, every ...
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5answers
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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 ...
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274 views

Tensor product of a number field $K$ and the $p$-adic integers

In a paper by J. Jones and D. Roberts (http://math.la.asu.edu/~jj/localfields/database.pdf) we are introduced to an isomorphism $K \otimes \mathbb{Q}_p \cong \prod\limits_{i=1}^g K_{p,i}$, where $K$ ...
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2answers
208 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 ...