0
votes
0answers
10 views

galois group of a biquadratic involving primes.

Let $f(x)=x^4-px^2+q \in \mathbb Q[x]$ be a polynomial with $p,q$ be distinct primes. Prove that $f$ it's irreducible over $\mathbb Q$. Prove that it's Galois group is the dihedral. I proved the ...
2
votes
0answers
36 views

Lucas' Theorem for $p$-adic integers?

Does Lucas' Theorem hold for $p$-adic integers? More specifically, does it specifically hold for the case that, given a $p$-adic integer: $$x = x_0 + x_1p + x_2p^2 + \cdots = \sum_{i=0}^\infty ...
1
vote
1answer
25 views

Something similar to Euler's theorem

If $p$, $q$ are not equal primes. $n=pq$, $\varphi(n) = (p − 1)(q − 1)$, $d = \gcd(p − 1, q − 1)$. Is it true that for any $a$ such that $\gcd(a, n) = 1$ holds $a^{\frac{\varphi(n)}{d}} \equiv 1 ...
-2
votes
0answers
28 views

The diophantine equation $A^2+B^2=C^2$ for integer-valued polynomials

How can I find the solutions to this diophantine equation in $\Bbb{Z}[X]$: $$A^2+B^2=C^2 \, ?$$ Here $A$, $B$, $C$ are polynomials.
0
votes
1answer
16 views

Equivalence for binary quadratic forms with positive square discriminant

I recently encountered an interesting proposition without proof: If $f(x,y)$ is a quadratic form whose discriminant is a non-zero perfect square, then $f(x,y)$ is equivalent to a form $a*x^{2} + ...
6
votes
0answers
43 views

Most elementary proof that a determinant is divisible by $m$

So a challenge problem states that you have an $n \times n$ matrix, where each entry is an integer between $0$ and $9$, and when each row is read as a base-10 number the number is divisible by a ...
2
votes
1answer
46 views

smallest element order $p$ in $\mathbb Z^*_{p^2}$

I would like to write an efficient algorithm to find the smallest element of order $p$ in $\mathbb Z^*_{p^2}$, where $p$ is a prime number. Therefore I calculate $a^{p-1} \pmod{p^2}$ for every ...
24
votes
1answer
518 views

Prove that both $x+y$ and $xy$ are rational, under some conditions

As a result of the answer I got for this question - Irrational solutions to some equations in two variables - I was wondering if the next statement is always true: Let $x,y$ be real, irrational ...
14
votes
1answer
238 views

Irrational solutions to some equations in two variables

The next statement is a conjecture of mine, so I dont know if it's true (though quite sure): Let $x,y$ be irrational numbers such that $x^4+y^4=1$. Prove (or disprove) that $x^5+y^5$ is irrational or ...
1
vote
0answers
38 views

Does number precede structure?

Initially, I was tempted to believe the notion of structure with respect to abstract algebra was more general than the notion of number in number theory. However, apparently there are a large number ...
0
votes
0answers
60 views

Exercise books in abstract algebra and number theory

I'm studying Herstein's Topics in algebra and Hardy&Wright's An introduction to the theory of numbers, and I was wondering if there are some exercise books (that is, books with solved problems and ...
0
votes
2answers
45 views

Sum of the elements of a cyclic subgroup

Let $G$ be a finite cyclic subgroup of the group of units of a commutative unital ring $R$. What is the sum of the elements of $G$, i.e. $$\sum_{x \in G}{x}?$$ [The answer is not difficult in the ...
0
votes
2answers
59 views

Can $\mathbb{Z}/n\mathbb{Z}$ (not $(\mathbb{Z}/n\mathbb{Z})^{\times}$) be a group under multiplication?

I was wondering why we usually say $\mathbb{Z}/n\mathbb{Z}$ is a group under addition and invent notation like $(\mathbb{Z}/n\mathbb{Z})^\times$ specifically for the multiplicative group modulo $n$. ...
2
votes
3answers
105 views

Prove or disprove $ p^{r+s}\mid q^{ke} - 1 \iff p^s \mid k$.

Let $p$ be an odd prime and $q$ be a power of prime. Suppose $e := \min\{\, e \in \mathbb{N} : p \mid q^e - 1 \,\}$ exists. Put $r := \nu_p(q^e - 1)$ (that is, $p^r \mid q^e - 1$ and $p^{r+1} \nmid ...
0
votes
1answer
38 views

Hilberts Theorem (norm group)

The theorem says the following: The map $N$ is a group homomorphisim from the multiplicative group of $\mathbb{Q}^{x}[i]$ to the multiplicative group of $\mathbb{Q}^{x}$ and has kernel $\lbrace ...
3
votes
1answer
66 views

Number theoretic proof that $n\mid\phi(a^n-1)$

While 'playing' with the multiplicative group of integers mod $n$, I noticed that $n\mid \phi(a^n-1)$. The proof is straightforward: $a \in \left ( \mathbb{Z}/(a^n-1)\mathbb{Z} \right ...
2
votes
1answer
76 views

Infinite primes of a number field

Let $K$ be a number field. I know that to each real and to each complex conjugate pair of embeddings of $K$ there corresponds exactly one prime (equivalence class of absolute values) of $K$. How do I ...
0
votes
1answer
30 views

Reverse the twisting of modular form

It is known that the twisting of the Fourier expansion of a modular forms by a Dirichlet character produce a modular form. ...
2
votes
2answers
31 views

Linear equation over $\mathbb{Z}/n\mathbb{Z}$

For given $a,b\in \mathbb{Z}/n\mathbb{Z}$ is there a criterion which allows one to determine whether there exists $x\in \mathbb{Z}/n\mathbb{Z}$ with $ax=b$?
0
votes
0answers
35 views

Applications of Hensel lemma

In the paper Roots of Polynomials Modulo Prime Powers by Bruce Dearden and Jerry Metzger, the authors state about the last table of the paper "Somewhat more complicated reasoning explains the ...
1
vote
0answers
30 views

Roots of polynomials modulo prime powers

I'm reading the paper Dearden, Bruce(1-ND); Metzger, Jerry, Roots of polynomials modulo prime powers. European J. Combin. 18 (1997), no. 6, 601–606. link: ...
-1
votes
1answer
53 views

Can I generalize this number theory result to finite chain rings?

I was working with $\mathbb{Z} \setminus p^m \mathbb{Z}$ and used the fact that $$\sum_{i=1}^{p-1} i^2 \equiv 0 \text{ mod p}.$$ Given an arbitrary finite chain ring $R$ with ideals $$\{ 0 \} = ...
0
votes
0answers
40 views

Subring of $\mathbb{Z}[i]$ and an infinite set $X$ such that $\exists x \forall y \in X \,\,x^2\mid y^2$ but $\forall x \forall y \,\,x \not\mid y$

This is a question derived from A subring of the ring of Gaussian integers such that $a^2 \mid b^2$ does not lead to $a\mid b$ in infinitely many such cases. Is there a subring $R$ of Gaussian ...
0
votes
1answer
49 views

A subring of the ring of Gaussian integers such that $a^2 \mid b^2$ does not lead to $a\mid b$ in infinitely many such cases

As the ring of Gaussian integers is a UFD, this means that $a^2 \mid b^2$ leads to $a\mid b$. Is there any subring of the ring of Gaussian integers with infinitely many elements such that ...
1
vote
1answer
44 views

What does “generate freely” mean?

Given a number field $K$ (i.e. $\mathbb Q\le\ K\le\mathbb C$, $[K:\mathbb Q]=n$), the relative number ring is $R=\mathbb A\cap K$, where $\mathbb A$ is the ring of the algebraic integers in $\mathbb ...
6
votes
4answers
163 views

When does $V/\operatorname{ker}(\phi)\simeq\phi(V)$ imply $V\simeq\operatorname{ker}(\phi)\oplus\phi(V)$?

When does $V/\operatorname{ker}(\phi)\simeq\phi(V)$ imply $V\simeq\operatorname{ker}(\phi)\oplus\phi(V)$? I wrote it down in an imprecise way on purpose. The notation above is the linear algebra one: ...
0
votes
0answers
19 views

There exists only a finite number of ideal classes in a number ring

Let $K$ be a number field (i.e. $\mathbb Q\le K\le\mathbb C$ s.t. $[K:\mathbb Q]=n$) and $R=\mathbb A\cap K$ the relative number ring. Calling $\Phi(R)$ the set of ideals of $R$, we define on it the ...
3
votes
0answers
40 views

Non-UFD that there exists set $X$ of any cardinality and $a$ that $xy$ is not divisible by $a^2$ for any $x,y \in X$ and $x^2$ is divisible by $a^2$

Let's say we want to construct a non-UFD that is a commutative ring that satisfies: There exists $a$ such that there exists a set $X$ of elements of finite cardinality $k$ such that for any $x,y \in ...
2
votes
1answer
76 views

An estimate involving gaps in a subsemigroup of $(\mathbb N,+)$

I think this question can be solved by a high school student, maybe there is some trick on it or I'm forgetting something. Before my question, some background is required: Definition: A ...
0
votes
3answers
44 views

Converting a polynomial ring to a numerical ring

One may be curious why one wishes to convert a polynomial ring to a numerical ring. But as one of the most natural number system is integers, and many properties of rings can be easily understood in ...
4
votes
2answers
87 views

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

Is there any non-UFD that is a commutative ring such that $a^2 \mid b^2$ does not always lead to $a\mid b$? It would be preferable if examples are something that does not involve ...
1
vote
3answers
60 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
146 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
48 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
55 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 ...
3
votes
0answers
43 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
40 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
133 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
39 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
83 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
58 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
71 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
36 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
38 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
53 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
69 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
48 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 ...