0
votes
0answers
2 views

Show that the HNF of the column vector $[a_{1},…,a_{n}]^{T}$ is exactly $[gcd(a_{1},…,a_{n}),0,…,0]^{T}$

HNF = Hermite Normal Form. I see why this is true by computing an example...and I know I need to use the Euclidean Algorithm...
3
votes
1answer
25 views

Ideal in Dedekind domain

Let $D$ be Dedekind domain and $I$ nonempty ideal in $D$. I have to show that there are only finitely many ideals $J$ in $D$ such that $I$ is contained in $J$. My first idea would be: assume that ...
0
votes
1answer
32 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 ...
0
votes
0answers
24 views

Ireland and Rosen, question 12.28 (unique factorization in prime ideals)

I'm stuck with the following exercise in Ireland and Rosen, chapter 12. Let $D$ be the ring of integers in a number field $F$. Suppose $(p)=P^2A$ for $p$ prime in $\Bbb Z$ and a prime ideal $P$. ...
2
votes
0answers
72 views

If $\alpha$ and $\beta$ are algebraic integers then the roots of $x^2+\alpha x+\beta$ are algebraic integers

(This question is a dupplicate from If $\alpha$ and $\beta$ are algebraic integers then any solution to $x^2+\alpha x + \beta = 0$ is also an algebraic integer.) I'm trying to solve this problem with ...
5
votes
1answer
154 views

The field of algebraic numbers in $\mathbb Q (a_1,\ldots, a_l)$ is finite over $\mathbb Q$

In the book 'Algebra IV: Infinite Groups, Linear Groups' by Kostrikin and Shafarevich, there is a sketch of a proof (on page 84) of a theorem by Schur. I'm struggling to understand the line: Since ...
0
votes
1answer
34 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 ...
1
vote
2answers
136 views

If $\alpha$ is an algebraic number, then prove that $\frac{2}{3}\alpha$ is also algebraic.

If $\alpha$ is an algebraic number, then prove that $\frac{2}{3}\alpha$ is also algebraic. Note: $\alpha \in \mathbb{R}$ I know that if $\alpha$ is algebraic, then there exists some $f(x) \in ...
2
votes
1answer
36 views

How do we extend the valuation on $K[x]$ to a valuation on $K(x)$?

Given that $v$ is a non-archimedean valuation on $K$, we can extend it to $|\cdot|:K[x]\to\mathbb{R}$ by $|a_0 + a_1x + \cdots +a_nx^n|=\max\left\{|a_1|,\ldots,|a_n|\right\}$. My question is how can ...
5
votes
2answers
128 views

$||x||=1$ in $K/\mathbb{Q}$ implies $x$ is a root of unity.

Let $K/\mathbb{Q}$ a finite (i.e. algebraic and finitely generated) extension. Let $x \in K$, such that $||x||=1$ for all normalized absolute values of $K$ but at most one. Then $x$ is a root of ...
2
votes
1answer
72 views

$Q_p(\zeta)$ where $\zeta$ is a $p$-th root of $1$.

I'm not looking for a full solution, only a hint please! Let $\zeta$ be a $p$-th root of unity in an algebraic closure of $Q_p$. Show that $Q_p(\zeta) = Q_p ((-p)^{\frac{1}{p-1}})$. Following a hint ...
2
votes
0answers
31 views

Lattice Bases of Prime (Ideal) Divisors

My question is: How can I find the prime (ideal) divisors of 2 and 3 in the ring of integers of $\mathbb Q[\sqrt{-14}]$ and $\mathbb Q[\sqrt{-23}]$? Here's what I have so far. I found that (2, ...
1
vote
1answer
116 views

How to show 2 cannot be totally ramified in $\mathbb{Q}(\sqrt{6},\sqrt{10})$?

I am trying to show that 2 cannot be totally ramified in $\mathbb{Q}(\sqrt{6},\sqrt{10})$. I know that it is totally ramified in $\mathbb{Q}(\sqrt{6})$ and $\mathbb{Q}(\sqrt{10})$ since ...
4
votes
3answers
174 views

Irreducibility of a particular polynomial

I've got this problem for my homework: find out whether the polynomial $$f(x)=x(x-1)(x-2)(x-3)(x-4) - a$$ is irreducible over the rationals, where $a$ is integer which is congruent to $3$ modulo $5$. ...
1
vote
0answers
44 views

Norms on $k(T)$ equivalent under automorphism

Let $k$ be a field and $K = k(T)$ the field of rational functions in one variable over $k$. Consider the two norms on $K$, which restricted to $k$ are trivial: $$\begin{aligned}& ...
0
votes
1answer
52 views

set theory, show sets are not of equal cardinality - check my proof

question from exam in set theory: let $M$ be the set of all real numbers x that satisfy: $cx^2+bx+a=0$ where $a,b,c \in Z$ (Meaning they are integers) and $c$ is not $0$. We will define $K = \{sm+t ...
3
votes
1answer
37 views

Why $(\alpha-1)^{-1}\le u^2$ where $u$ is a fundamental unit in $\mathbb{Z}[\alpha]$ and $\alpha=2^{1/3}$?

Given $\alpha = 2^{1/3},$ I want to show that $\beta = (\alpha-1)^{-1}$ is a unit in $\mathbb{Z}[\alpha]$ and is between 1 and $u^2$, where $u$ is a fundamental unit in $\mathbb{Z}[\alpha]$. I see ...
4
votes
1answer
42 views

Discrete valuation on a field - equivalent statements

I have a question and I am stuck, although it should not be too difficult. We consider $K$ a field, $v$ a discrete valuation on $K$ and $O=\{x \in K:v(x)\geq 0\}$ the valuation ring of $v$. Let ...
0
votes
2answers
113 views

irreducible polynomial in $Z_p [x]$

Let $p$ be a prime integer. prove that the following are equivalent (a) $p$ is a prime in $Z[i]$ (b) $x^2 +1$ is irreducible in $Z_p [x]$ What I know is that if p is a prime in $Z[i]$, $p$ is ...
1
vote
2answers
81 views

Decomposition of an ideal as a product of two ideals

How to show $$5\mathbb{Z}[\sqrt[3]{2}] = (5, \sqrt[3]{2}+2)(5, (\sqrt[3]{2})^2+3\sqrt[3]{2}-1).$$ Firstly, I think that I can say that $$(5, \sqrt[3]{2}+2)(5, (\sqrt[3]{2})^2+3\sqrt[3]{2}-1)= ...
0
votes
1answer
62 views

work out the value of a - b from the identity $ax+18=2(x-b)$

How do I solve the following question? You are given the algebraic identity: $ax+18=2(x-b)$ Work out the values of $a-b$
1
vote
0answers
80 views

Showing $L/K$ is unramified for almost all primes when $L=K(\sqrt[n]{a}), a\in A$ and $(a) = I^n$

This question comes from Lorenzini's book "An Invitation to Arithmetic Geometry." It's question 26b on page 129. We're given that $A$ is a Dedekind domain with field of fractions $K$, and that ...
3
votes
1answer
222 views

Why is $Q(\sqrt{-1}, \sqrt{-5})$ unramified over $Q(\sqrt{-5})$?

I'm working on a problem in Lorenzini's book "An Invitation to Arithmetic Geometry" which asks to show that if $L = Q(\sqrt{-5}, \sqrt{-1})$ and $K = Q(\sqrt{-5})$, then the ring of integers of $L$ is ...
3
votes
1answer
50 views

Norm of an element coprime to a prime algebraic number

Let $\pi:=1+\sqrt{3}$ be an element of $\mathbb{Z}[\sqrt{3}]$. I have proved that $\pi$ is a prime number in $\mathbb{Z}[\sqrt{3}]$. Now let $\alpha$ be another element in $\mathbb{Z}[\sqrt{3}]$, such ...
0
votes
1answer
86 views

Prove that the subdomain $\mathbb{Z}+ 7\mathbb{Z}\sqrt{2}$ of the Euclidean domain $\mathbb{Z}+\mathbb{Z}\sqrt{2}$ is not Euclidean

I need help to prove that the subdomain $\mathbb{Z}+ 7\mathbb{Z}\sqrt{2}$ of the Euclidean domain $\mathbb{Z}+\mathbb{Z}\sqrt{2}$ is not Euclidean. I have been using the Alaca & Williams book, ...
2
votes
1answer
42 views

Proof involving characters

I'm self studying from a Classical Introduction to Modern Number Theory by Ireleand and Rosen. In the exercises for the chapter on Gauss and Jacobi sums I came across this question. Let $\chi$ be a ...
6
votes
0answers
191 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 ...
5
votes
2answers
228 views

Is $\sin(1)$ algebraic over $\mathbb{Q}$?

Is $\sin(1)$ algebraic over $\mathbb{Q}$? At the moment I have no idea how to proceed. Could you tell me how to solve it?
7
votes
1answer
218 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 ...
2
votes
1answer
81 views

How to find $\sup(\{|x-y|_p : x,y\in B(0;r)\})$

Just to clarify the notation and the question: Working in p-adic space $\mathbb{Q}_p$, we have the norm $|x|_p=p^{-ord_p(x)}$ and we define the metric over this space as $d(x,y)=|x-y|_p$. We are ...
4
votes
2answers
92 views

Proving a factorization of ideals in a Dedekind Domain

Let $R=\mathbb{Z}[\sqrt{-13}]$. Let $p$ be a prime integer, $p\neq 2,13$ and suppose that $p$ divides an integer of the form $a^2+13b^2$, where $a$ and $b$ are in $\mathbb{Z}$ and are coprime. Let ...
3
votes
2answers
144 views

Factoring the ideal $(8)$ into a product of prime ideals in $\mathbb{Q}(\sqrt{-7})$

I am trying to factor the ideal $(8)$ into a product of prime ideals in $\mathbb{Q}(\sqrt{-7})$. I am not exactly sure how to go about doing this, and I feel I am missing some theory in the ...
4
votes
1answer
165 views

Solving $x^2+19=y^5$

I was given several exercises and there is a particular one, I am not able to solve. Let it be given that $Pic(\mathbb{Z}[\sqrt{−19}])$ is a finite group of order $3$. Use this to find all integral ...
5
votes
2answers
83 views

Number fields and Integral Closure

I am trying to solve the following problem: Consider the fields $K=\mathbb{Q}[\omega]$, where $\omega^2+\omega+1=0$, and $L=K(x)[y]$ where $y^3=1+x^2$, and the rings $R=K[x]$, and ...
1
vote
1answer
61 views

An exercise on lattices

How to show that $L = \{(x,y,x)\in\mathbb{Z}^3 : 2x + 3y + 4z\equiv 0\pmod 7\}$ is a lattice?
0
votes
1answer
167 views

Norm of an ideal

Would someone attempt to give me a simple explanation of how to compute the norm of an ideal. I like the definition |$O/a$| but find it difficult to apply. Perhaps the "determinant version" lends ...
2
votes
1answer
78 views

Algebraic Number Theory: orders and fraction fields

I have a question in Algebraic Number Theory. I somehow do not know how to proceed. Let $O\subseteq \mathbb{Z}_K$ be subring where $K$ is a number field. TFAE: $\quad$ i) $O$ is an order of $K$ ...
2
votes
3answers
116 views

Proving that $ 2 + 3\sqrt{-2} $ is reducible in $ \mathbb{Z}[\sqrt{-2}] $

Prove that $ 2 + 3\sqrt{-2} $ is irreducible in $ \mathbb{Z}[\sqrt{-2}] $ So far, I have let $ 2 + 3\sqrt{-2} = (a + b\sqrt{-2})(c+ d\sqrt{-2}) $ I then took the norm and got $\mathbf{N}(2 + ...
-2
votes
2answers
366 views

How many prime ideals does $\mathbb Q[x]/(x^m -1)$ have? (multiple choice)

Let $m$ be a positive integer, and $a_m$ denote number of distinct prime ideals of $\mathbb Q[x]/(x^m -1)$. Then which of the following are true? $a_4=2$ $a_4=3$ $a_5=2$ $a_5=3$
2
votes
1answer
260 views

How to find all the ideals of a given norm?

I am working on a question: Find all the ideals of norm $10$ in $\mathcal{O}_K$ where $K=\mathbb{Q}(\sqrt{35})$. I am given the hint: Observe that $(2)=(2,\alpha)^2, (5)=(5,\alpha)^2, ...
16
votes
3answers
276 views

How to prove summation, multiplication, subtraction of two roots of $1+x+\frac{x^2}{2!}+\cdots+\frac{x^p}{p!}=0$ aren't rationals?

Assume $a$, $b$ are distinct roots of the following equation: $$1+x+\frac{x^2}{2!}+\cdots+\frac{x^p}{p!}=0,$$ where $p$ is a prime number and $p \gt 2$. How to prove that $ab$, $a+b$, $a-b$ are not ...
4
votes
4answers
161 views

Algebraic Number Theory - Lemma for Fermat's Equation with $n=3$

I have to prove the following, in my notes it is lemma before Fermat's Equation, case $n=3$. I was able to prove everything up to the last two points: Let $\zeta=e^{(\frac{2\pi i}{3})}$. Consider ...
-2
votes
1answer
146 views

Show that $p$ and $q$ are not principal, but that $p^2$, $pq$ and $q^2$ are.

Let $K$ be the field $\mathbb Q(\sqrt{−15})$, let $R = \mathcal{O}_K$ be the ring of integers of $K$. Let $\alpha= \frac{-1+\sqrt{-15}}{2}$ and consider the prime ideals $p = (2,α)$ and $q = (17,α + ...
13
votes
2answers
832 views

On the ring of integers of a compositum of number fields

This is Daniel A. Marcus, Number Fields, Exercise 2.29 If anyone can help with this problem, I'd greatly appreciate it. Let $K$ be the biquadratic field $\mathbb Q[\sqrt{m}, \sqrt{n}] = \{a + ...
4
votes
1answer
363 views

Find an integral basis

$K=\mathbb{Q}(\alpha)$ where $\alpha$ has minimal polynomial $X^3-X-4$. Find an integral basis for $K$. I have calculated the discriminant of the minimal polynomial is $-2^2 \times 107$, so the ring ...
2
votes
1answer
309 views

Ring of integers of a cubic number field

Let $K=\mathbb{Q}(a)$ where $a^3=d$ where $d\neq 0, \pm 1$ is a square free integer. Show that $\Delta (1, a, a^2)=-27d^2$. By calculating the traces of $\theta, a\theta, a^2\theta$ where ...
2
votes
2answers
100 views

How can we determine the factorisation of $2\mathcal{O}_L$?

Let $L =\Bbb{Q}(\sqrt{2},\sqrt{-3})$ and $ K = \Bbb{Q}(\sqrt{2})$. Now the prime $2$ is totally ramified in two of the three subfields of $L$ and from ramification theory we know that $2$ splits as ...
5
votes
1answer
163 views

Typo in Marcus' $\textit{Number Fields}$?

I am doing Problem 5.10 of Marcus where it is given that $m$ is a square-free negative integer and that $\mathcal{O}_K$ is a PID where $K = \Bbb{Q}(\sqrt{m})$. Now in part (b) of this problem he ...
9
votes
2answers
410 views

Exact power of $p$ that divides the discriminant of an algebraic number field

I am doing Marcus problem 21 (b) of chapter 3. The setup for this problem is given in problem 20: Setup: Let $L/K$ be a finite extension of algebraic number fields. Write $R = \mathcal{O}_K$ ...
6
votes
2answers
126 views

Alternative proof that $[\Bbb{Q}(\zeta_n) : \Bbb{Q} ]= \varphi(n)$ uses circular reasoning?

I am doing exercise 3.24 of Marcus which is the following. Let $L,K$ be number fields with $L/K$ a finite extension (of degree $[L:K] = n$) with $R = \mathcal{O}_K$ and $S = \mathcal{O}_L$. ...