Questions related to the algebraic structure of algebraic integers

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Integers of the form $x^2+2y^2$.

I'm stuck in the following problem: prove an integer $n$ is of the form $x^2+2y^2$ if and only if every prime divisor $p$ of $n$ that is congruent to $5$ or $7\bmod8$ appears with an even exponent. I ...
8
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2answers
497 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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1answer
461 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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2answers
519 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, ...
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1answer
339 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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3answers
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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 ...
8
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2answers
338 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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3answers
186 views

Ring of integers of cubic number field

I want to show that the ring of integers of the cubic number field $K = \mathbb Q(\alpha)$, where $\alpha$ is a root of $f = X^3 - X - 2$, is equal to $\mathbb Z[\alpha]$. $(1, \alpha, \alpha^2)$ ...
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1answer
389 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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3answers
125 views

How to prove there are no solutions to $a^2 - 223 b^2 = -3$.

As the title suggests, I'm trying to prove that there are no solutions to $a^2 - 223b^2 = -3$ (with $a,b\in \mathbb{Z}$). Ordinarily, taking both sides $\mod n$ for some clever choice of $n$ proves ...
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3answers
396 views

Prove that $2$, $3$, $1+ \sqrt{-5}$, and $1-\sqrt{-5}$ are irreducible in $\mathbb{Z}[\sqrt{-5}]$.

So the Norm for an element $\alpha = a + b\sqrt{-5}$ in $\mathbb{Z}[\sqrt{-5}]$ is defined as $N(\alpha) = a^2 + 5b^2$ and so i argue by contradiction assume there exists $\alpha$ such that $N(\alpha) ...
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3answers
209 views

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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1answer
734 views

How to arrive at Ramanujan nested radical identity

I've come across a very curious Ramanujan identity $$\sqrt[6]{7 \sqrt[3]{20} - 19} = \sqrt[3]{\frac{5}{3}} - \sqrt[3]{\frac{2}{3}}$$ You could probably prove this by taking the 6th power of both ...
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1answer
1k 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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2answers
651 views

On the class group of an imaginary quadratic number field

Let $d < 0$ be a square-free integer and let $p_{1},\ldots p_{r}$ be the prime divisors of $d$. Let $K := \mathbb{Q}[\sqrt{d}]$ and consider $P_{i} := (p_{i}, \sqrt{d}) \subset \mathcal{O}_{K}$. ...
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2answers
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Why if $\alpha$ is a root of $x^3-5x^2+2$, then $\mathcal O_{\Bbb Q[\alpha]}=\mathbb Z[\alpha]$?

I'm trying to show that if $\alpha$ is a root of the polynomial $x^3-5x^2+2$ and $K=\Bbb Q[\alpha]$, then $\mathcal O_K=\Bbb Z[\alpha]$. This is homework, and one of the previous exercises asks to ...
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2answers
190 views

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
196 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
122 views

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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1answer
374 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 ...
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1answer
84 views

Zeta function, $\mathbb{F}_5[T, \sqrt{T(T-1)(T+1)}]$

Let $A = \mathbb{F}_5[T, \sqrt{T(T-1)(T+1)}]$. My question is, what is the easiest way to see that$$\zeta_A(s) = {{1 + 2 \cdot 5^{-s} + 5^{1 - 2s}}\over{1 - 5^{1 - s}}}?$$Much thanks in advance. ...
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2answers
90 views

Irreducibility of cyclotomic polynomials over number fields

Let $K$ be a number field, i.e., a finite extension of $\mathbb{Q}$. For a positive integer $n$, let $\Phi_n(X)$ denote the $n$-th cyclotomic polynomial. Is it possible to say that there exist at ...
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1answer
120 views

How to test if a given elliptic curve has complex multiplication

Is there a general, reasonably easy to understand, algorithm for testing whether an elliptic curve has CM? For example, consider the curve $y^2=x^3+\frac{27}{1727}x+\frac{54}{1727}$ This has ...
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3answers
381 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 ...
8
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1answer
211 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
138 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
270 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 ...
8
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2answers
140 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
139 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
160 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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1answer
153 views

What is the Galois group of $\mathbb{Q}_p[\zeta] / \mathbb{Q}_p$, where $\zeta$ is a $p^r$th root of unity?

Let $\zeta$ be a $p^r$-th root of unity and $\mathbb{Q}_p$ the p-adic numbers. I would like to know the Galois automorphisms of $\mathbb{Q}_p[\zeta] / \mathbb{Q}_p$. I already know, the degree of ...
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1answer
191 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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2answers
624 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
402 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 ...
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3answers
320 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
74 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 ...
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1answer
166 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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1answer
492 views

Ramanujan-Nagell Theorem Proof Question

I'm currently working through Stewart and Tall's Algebraic Number Theory. In particular, section 4.9 of this book provides a proof of the Ramanujan-Nagell Theorem, which states that the only integer ...
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2answers
299 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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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
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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 ...
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Specializations of elementary symmetric polynomials

Let $\mathcal{S}_{x}=\{x_{1,},x_{2},\ldots x_{n}\}$ be a set of $n$ indeterminates. The $h^{th}$elementary symmetric polynomial is the sum of all monomials with $h$ factors \begin{eqnarray*} ...
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Coefficients in expansion of $(\sqrt[3]{2} - 1)^m$

In trying to solve $a^3 - 2b^3 = 1$ over the integers I came across the need to answer the question: when does $(1+ \sqrt[3]{2} + \sqrt[3]{2}^2)^n$ have no $\sqrt[3]{2}^2$ term in it's expansion (in ...
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1answer
263 views

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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666 views

Compute the Unit and Class Number of a pure cubic field $\mathbb{Q}(\sqrt[3]{6})$

Find a unit in $\mathbb{Q}(\sqrt[3]{6})$ and show that this field has class number $h=1$. I am done with the first part which is relatively simple: Suppose that $\varepsilon$ is a unit in ...
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0answers
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notation for ramification index and inertial degree

For a prime $Q$ lying over a prime $P$, I have seen the ramification index of $Q$ over $P$ denoted by $e(Q|P)$ and the inertial degree of $Q$ over $P$ by $f(Q|P)$. What is the origin of the ...
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226 views

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 ...
8
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1answer
225 views

Adelic/Idelic method for number fields

I'm searching for a place where is developed all the machinery of adeles and ideles of a number field; the functional equation for the zeta functions is gained by idelic integration, and it's seen the ...
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4answers
1k views

Lack of unique factorization of ideals

I'm aware of the result that integral domains admit unique factorization of ideals iff they are Dedekind domains. It's clear that $\mathbb{Z}[\sqrt{-3}]$ is not a Dedekind domain, as it is not ...
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5answers
614 views

Give an example of a UFD having a subring which is not a UFD.

Give an example of a UFD having a subring which is not a UFD. I thought of $\mathbb{Z}[\sqrt{2},\sqrt{3}]$. Could you please explain my question. I am trying grasp the concepts, need help.