0
votes
2answers
17 views

A non-UFD such that $a^2 | b^2$ does not lead to $a|b$

Is there any non-UFD that is a commutative ring such that $a^2 | b^2$ does not always lead to $a|b$? ($|$ refers to divisible by) It would be preferable if examples are something that does not ...
0
votes
2answers
41 views

A number system that is not unique factorization domain

Can anyone present a number system that is not unique factorization domain and is a commutative ring? So I want the case that does not involve polynomials/monomials or some trivial cases.
2
votes
0answers
141 views

$e=1$ in Theorem 30 from Marcus book “number fields”

Theorem 30 in Marcus book states that, if $p\in\mathbb Z$ is an odd prime and $q$ is a prime $\neq p$, then, fixing $d$ as a divisor of $p-1$ we have that $q$ is a $d$-th power $\operatorname{mod}q$ ...
1
vote
2answers
46 views

$a^2-b^2 = k$, $ab = l$ for fixed integer $k,l$ when $a,b$ are both integers

Let us fix integers $k,l$. Let all numbers be integers. Now we want integer $a,b$ to satisfy: $$a^2-b^2 = k, \,\,\,2ab = l.$$ We want to maximize the number of possible $(a,b)$. In order to do ...
2
votes
2answers
52 views

Can there ever be infinite number of tuples of $(a,b,c,d)$ such that $ac-bd = k$ and $ad+bc = l$ for fixed $k,l$?

Suppose, for now, that all numbers are real numbers. Let us fix numbers $k,l$. Then can there ever be infinite number of tuples of $(a,b,c,d)$ such that $ac-bd =k$, $ad+bc = l$ for some $k$ and $l$? ...
1
vote
1answer
16 views

$p\in\mathbb Z$ ramified in $R\Rightarrow p|\operatorname{disc}(R)$

The one in the title is Theorem 24, page 72 in the Marcus book "Number Fields". I have a problem with a detail in the last part of the proof. We have $\mathbb Q\le K$ and $L$ is a normal extension ...
-1
votes
0answers
22 views

A commutative ring that satisfies $(x=0) \rightarrow ((\forall z \forall y \,\, x = yzz) \vee (\forall y \forall z \,\, x=yz+zy))$

Can anyone give an example of a commutative ring that satisfies $(x=0) \rightarrow ((\forall z \forall y \,\, x = yzz) \vee (\forall y \forall z \,\, x=yz+zy))$ and $\forall x \,\, x \cdot x = 0$? ...
3
votes
0answers
42 views

How can I compute |R/I| where $R\simeq\mathbb Z^n$ and $I\unlhd R$?

A number field $K$ is a field $\mathbb Q\le K\le\mathbb C$ of finite degree over $\mathbb Q$, say $n$. Call $\mathbb A$ the ring of algebraic integers of $\mathbb C$; an algebraic integer is an ...
0
votes
1answer
36 views

Detail in Theorem 12 pag 33, from Marcus book “Number Field”

Let $K, L$ be number fields (i.e. subfields of $\mathbb C$ of finite degree over $\mathbb Q$) of degree $m, n$ over $\mathbb Q$ respectively and assume $[KL:\mathbb Q]=nm$. Consider $KL$ to be the ...
0
votes
4answers
121 views

How do you go about finding a 12 digit prime number?

How do you go about finding a 12 digit prime number?
2
votes
3answers
69 views

Finding the square root $s$ of 1293 modulo 3337.

If $3337 = 47 \cdot 71$, how do you find the square root $s$ of $1293 \pmod { 3337}$ (where $0 < s < 3337$). I understand that $m = 3337 = p \cdot q$ and $p=47$ and $q=71$, but not sure where ...
1
vote
1answer
32 views

Norm of $1-\omega$ where $\omega=e^{\frac{2\pi i}p}$

We are working in the number field $\mathbb Q[\omega]$, where $\omega=e^{\frac{2\pi i}p}$, $p$ prime. This number field had degree $p-1$ over $\mathbb Q$ hence there are $p-1$ embeddings of $\mathbb ...
5
votes
3answers
79 views

Why $f\colon \mathbb{Z}_n^\times \to \mathbb{Z}_m^\times$ is surjective?

If $m|n$. Why the map $f\colon \mathbb{Z}_n^\times \to \mathbb{Z}_m^\times$ given by $a \mod{n}\mapsto a \mod m$ is a surjective homomorphism of groups? Attempt: I proved it is well a well defined ...
7
votes
1answer
51 views

An extension of an algebraic number field which makes an integral ideal $I$, a principal ideal

I want to show that, given an ideal $I \subseteq \mathcal O_K$ (where $K/\mathbb Q$ is an algebraic number field), there is a finite extension $K'/K$ such that, $I\mathcal O_{K'}$ becomes a principal ...
4
votes
1answer
65 views

Milne's Galois Theory Example

The following example is drawn from Milne's Galois Theory notes, p.42 (http://www.jmilne.org/math/CourseNotes/FT.pdf) We study the extension $\mathbb{Q}[\zeta]/\mathbb{Q}$ where $\zeta=e^{2\pi i/7}.$ ...
1
vote
3answers
33 views

Ideals of the residual classes $\mathbb Z_n$

Let $n$ be a positive integer and considere the ring $\mathbb Z_n$ of the residual classes modulo $n$. My intuition tells me that there is no two distinct ideals of $\mathbb Z_n$ with the same number ...
0
votes
2answers
35 views

product of comprime numbers and UFD

It is well-known that if a product of coprime numbers is a perfect square, so are the numbers. The proof depends on fundamental theorem of arithmetic, and this implies that in a UFD, if ab is a ...
1
vote
1answer
46 views

When Is there no Local Power Integral Basis?

Let $A$ be a Dedekind domain, $K, L, B$ the usual designations of $A$'s quotient field, a finite separable extension of $K$, and integral closure of $A$ in $L$ respectively. If $\alpha \in B$ ...
1
vote
1answer
22 views

Extensions of nonarchimedean valuations

This is a question from Janusz 'Algebraic Number Theory'. Let $R$ be a DVR with maximal ideal $\mathfrak p=\pi R$. Let $K$ be the quotient field of $R$ and $\mid\cdot\mid_{\mathfrak p}$ the ...
1
vote
3answers
67 views

Additive identity in $\mathbb{Z}_{7}$

The problem consists of two parts: 1)Prove that for some integer $m,$ $3^{m} = 1$ in the integers modulo 7, $\mathbb{Z}_{7}$, which i have done. The second part asks to use this fact to deduce that ...
5
votes
0answers
45 views

Involutions of the second type in a division algebra

I'm trying to figure out some details about involutions of division algebra, thought maybe someone here might have a better insight. Let $k$ be a $p$-adic or number field, and let ...
3
votes
1answer
39 views

Showing two numbers are relatively prime in number fields

Solve $x^3-2=y^2$ in integers. The standard way to solve this problem is to consider the arithmetic of the ring of algebraic integers $\mathcal{O}_{\mathbb{Q}(\sqrt{-2})}$ and to show that ...
23
votes
1answer
248 views

Does there exist two non-constant polynomials $f(x),g(x)\in\mathbb Z[x]$ such that for all integers $m,n$, gcd$(f(m),g(n))=1$?

Does there exist two non-constant polynomials $f(x),g(x)\in\mathbb Z[x]$ such that for all integers $m,n$, gcd$(f(m),g(n))=1$? I think there are no such polynomials, but how to prove?
5
votes
3answers
49 views

Let $\mathbb Q\le K$ finite extension of fields $\alpha\in K$. Then $N(\alpha)=\pm1\Leftrightarrow\alpha$ is a unit

Let $\mathbb Q\le K$ (where $K\le\mathbb C$) be a finite extension (say $|K:\mathbb Q|=n$) of fields and let $\alpha\in K$ be an algebraic integer, i.e. $\alpha$ is a root of a monic polynomial over ...
5
votes
1answer
99 views

Units of $\mathbb{Z}[\sqrt[4]2]$

How would one compute the units in $\mathbb{Z}[\sqrt[4]2]$? According to one source, it can be shown that the fundamental units are $1 + \sqrt[4]2$ and $1 + \sqrt{2}$, but it does not specify the ...
3
votes
1answer
68 views

Determine Units of a Ring $\mathbb{Z}[\alpha]$

I am trying to determine the units of $Z[\alpha]$ where $\alpha$ satisfies the monic polynomial $\alpha^4+\alpha^3+\alpha^2+\alpha+1$. I found $Z[\alpha] := \lbrace a+b\alpha+c\alpha^2+d\alpha^3\;|\; ...
3
votes
1answer
44 views

tensor product writing of $\mathbb{C} / (2\pi i)^n \mathbb{Q}$

Can someone explain to me why $\mathbb{C}/(2\pi i)^n \mathbb{Q}$ is isomorphic to the tensor product $(2\pi i)^{n-1} \mathbb{C}^{\times} \otimes_{\mathbb{Z}} \mathbb{Q}$? I don't understand well the ...
4
votes
2answers
97 views

The prime elements of the ring $ \mathbb{Z} \! \left[ \dfrac{i \sqrt{3} - 1}{2} \right] $.

I have to study the prime elements of the ring $ \mathbb{Z} \! \left[ \dfrac{i \sqrt{3} - 1}{2} \right] $. For the moment, I cannot find the general form of such elements. Can you help me? Thanks! :) ...
0
votes
1answer
41 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 , ...
0
votes
1answer
34 views

Modulation and translation properties of DFT

Consider the discrete fourier transform over a finite field $GF(q)$. Let also $\omega$$\in$$GF(q)$ be an element of order $n$ and which is an $n$-th root of unity. Definition 1. Let $v$ = ($v_0$, ...
1
vote
2answers
48 views

Show that it doesn't exist any of natural number $ n = 4m + 3$ that $ n= x^2+y^2 $ for any natural x and y [duplicate]

Show that it doesn't exist any of natural number $ n = 4m + 3$ that $ n= x^2+y^2 $ for any natural x and y Show that every prime number in form $ p=4m+1 $ could be showed as $ p = x^2+y^2$ (x and y ...
3
votes
2answers
97 views

Is there any usage of this sum formula?

Let $\phi$ be the Euler phi function, then $$n=\sum_{d|n}\phi(d)$$ It is one of the most famous formulas in number theory. It can be generalized by using groups. Let $G$ be a group of order $n$ and ...
1
vote
1answer
55 views

Number of solutions of some congruence equations.

How many $[u]\in(\mathbf{Z}/ab\mathbf{Z})^\ast$ satisfy the equations $u\equiv 1 \bmod \ a$, $u\equiv 1 \bmod \ b$? I somehow believe that the answer might be $(a,b)$. Is this actually true? Is the ...
2
votes
1answer
74 views

An exact sequence of unit groups

In the answer of K. Conrad to this question, he mentions a "nice 4-term short exact sequence of abelian groups (involving units groups mod a, mod b, and mod ab)" proving the product formula for ...
1
vote
1answer
21 views

Does $a^{\phi(p^k)/2} \equiv -1 \pmod {p^k}$ imply $a$ is a primitive root modulo $p$, where $p$ is an odd prime and $k$ an integer $\ge 2$?

Suppose $a$ is a primitive root $\pmod p$. Does $$a^{\phi(p^k)/2} \equiv -1 \pmod {p^k}$$ imply $a$ is a primitive root modulo $p^k$, where $p$ is an odd prime and $k$ an integer $\ge 2$ ? I've been ...
1
vote
1answer
41 views

Prove $n>1$ is prime $\iff \alpha^{n-1} = 1 \ \forall \alpha \in \mathbb Z_n\setminus\{[0]\}$.

Prove $n>1$ is prime $\iff \alpha^{n-1} = 1 \ \forall \alpha \in \mathbb Z_n\setminus \{[0]\}$. I have proven $\Rightarrow$ which is an immediate consequence of Euler's theorem, however I ...
2
votes
3answers
122 views

Questions for first year students at the University. [closed]

I will help teach in a introductory class in mathematics for engineers in applied math at the University. Anyone have any good and cool favorite questions or know where I can find some? Anything is ...
0
votes
2answers
85 views

Closure of Integers under multiplication and rational exponentiation

What is the closure of the Integers under a finite number of multiplications and rational exponentiations? For example, $3^{1/2}$, $i = -1^{1/2}$, and $\frac{-1+i \sqrt(3)}{2} = 1^{1/3}$ all in this ...
2
votes
1answer
49 views

Proof of Pocklingtons theorem: why do we have $\text{ord}_n(a) \mid (q-1)$ but $\text{ord}_n(a) \nmid (n-1)/p_i$?

Proof of Pocklingtons theorem: why do we have $\text{ord}_n(a) \mid (q-1)$ but $\text{ord}_n(a) \nmid (n-1)/p_i$ ? And why does $p_i^{\alpha_i} \mid \text{ord}_n(a)$ ? I see that ...
5
votes
1answer
65 views

Using Plancherel's Theorem to Prove the Gauss Sum

I'm interested in proving the following: Where $p$ is an odd prime and $z$ is a primitive $p$th root of unity, we let $Q(p)=\sum^{p−1}_{k=0}z^{k^2}$. Prove: $|Q(p)|=\sqrt{p}$. Specifically, I want ...
1
vote
3answers
105 views

Proof of k'th power theorem: $x^k \equiv a \pmod n$ has a solution $\iff$ $a^{\phi(n)/\gcd(k, \phi(n))} \equiv 1 \pmod n$.

Proof of k'th power theorem: $x^k \equiv a \pmod n$ has a solution $\iff$ $a^{\phi(n)/\gcd(k, \phi(n))} \equiv 1 \pmod n$, where $n$ has a primitive root. I have proven the following theorem ...
2
votes
1answer
101 views

Euclid's Lemma for Euclidean Ring.

Question: If $R$ is a euclidean ring and $\pi\in R$ is irreducible, prove that $\pi\mid\alpha\beta$ implies $\pi\mid\alpha$ or $\pi\mid\beta$. A solution is to prove all euclidean rings are PIDs, ...
2
votes
0answers
30 views

automorphism of $\Bbb{Q_p}$closed [duplicate]

How to show that there does not exist an automorphism of $\Bbb{Q_p}$ except identity. Please help. Is any automorphism of $\Bbb{Q_p}$ already continuous.?
1
vote
1answer
85 views

Question on complete discrete valuation field.

Let $F$ be a complete discrete valuation field and $f(X) = X^n + a_{n-1}X^{n-1} +\cdots+ a_0$ is an irreducible polynomial over $F$. How to show that a) $ v(a_0) > 0$ implies $v(a_i) > 0$ for ...
3
votes
2answers
60 views

How can I prove an ideal is a product of two irreducible ones

I'm trying to solve this question: I have a guess that $(6+\sqrt{11})=(2,4+\sqrt{11})(2,-3\sqrt{11})$ using some formulas in this book page 48. However I couldn't verify if the multiplication of ...
2
votes
2answers
126 views

Silly mistake in this number theory book

My question is very easily to be solved (at least I hope so) I think this book has a mistake: When I calculate I get $b_3\equiv -2 (\mod{2})$ which implies $b_3=0$, am I right? Another question, ...
0
votes
0answers
23 views

Why does this theorem provide a neccesary and sufficient condition for a $2 \times 2$ linear system of congruences to have a unique solution?

Why does the following theorem both provide a neccesary and sufficient condition for a $2 \times 2$ linear system of congruences to have a unique solution ? I see that $\gcd((ad-bc),m) = 1$ is a ...
6
votes
1answer
76 views

Is the extension Galois if $\mathrm{Aut}(K)$ acts transitively on the non-ramified prime ideals?

Let $K/\mathbb Q$ be a finite extension such that $\mathrm{Aut}(K)$ acts transitively on the prime ideals that are not ramified above the same prime $p\in\mathbb N$. Is $K$ Galois? Thanks in advance. ...
2
votes
1answer
87 views

Degrees of factors of polynomial $f(x)=x^q-(ax^2+bx+c)\in \mathbb{F}_q[x]$

I’m facing the problem of factoring polynomials of type $f(x)=x^q-(ax^2+bx+c)\in \mathbb{F}_q[x]$ and the degrees of factors seem to be quite special. For example, according to my experimental results ...
0
votes
0answers
15 views

$m^2 = \prod_{i=1}^k p_i^{2e_i} \equiv 1 + \sum_{i=1}^k e_i (p_i^2-1) (\mod 64)$ imply $(m^2 - 1)/8 \equiv \sum_{i=1}^k e_i (p_i^2-1)/8 (\mod 2)$?

Let $m$ be an odd positive integer with canonical decomposition $\prod_{i=1}^k p_i^{e_i}$. I know $$m^2 = \prod_{i=1}^k p_i^{2e_i} \equiv \prod_{i=1}^k [1+e_i(p_i^2-1) \equiv 1 + \sum_{i=1}^k e_i ...