# Tagged Questions

24 views

### Automorphism of $ℚ[\sqrt{d}]$

I came across this line from class note: $\mathrm{Aut}(ℚ[\sqrt{d}]) = \{1, σ\}$. I understand all the terms separately: “$\mathrm{Aut}$” stands for automorphism which means bijective homomorphism into ...
53 views

### Find the inverse of the equivalence class

I have to check if the equivalence class has an inverse(without calculations). If yes, I have to find it. $$[7] \in \mathbb{Z}_{36}$$ We know that $[a] \in \mathbb{Z}_m$ has an inverse ...
52 views

### Chinese Remainder Theorem stuff (but more advanced, certainly derived from)

First, let $m$,$n$ be coprime. Suppose we want to find: $x\equiv y\text{ mod }m$ $x\equiv z\text{ mod }n$ As m,n are coprime $\exists a,b\in\mathbb{Z}:am+bn=1$ (GCD=1 basically) Then, for ...
14 views

### Berkovich analytifications and non-archimedean geometry of Transseries

In Transseries and Real Differential Algebra by Joris van der Hoeven it is said that Transseries admit a rich non-Archimedean geometry (somewhere on page 13), but since the book isn't about that, ...
29 views

36 views

### Primality of a binary number

I'm interested in finding out if there is a way to detect if a given binary number is prime. I do not which to convert it to base 10 then use some primality test. Does anyone know it there's a pattern ...
90 views

54 views

### Prime Splits Completely in Every Intermediate Field

Suppose I have a finite field extension of number fields (finite field extensions over $\mathbb{Q}$), say $K\subset L$. Say $P$ is a prime in the number ring contained in $K$ such that $P$ splits ...
47 views

### List of Primes in UFD

Are there websites/databases containing lists ordered by norm of prime/irreducible elements in domains like $\mathbb{Z}[\sqrt{-2}]$ and $\mathbb{Z}\left[\frac{-1+\sqrt{-3}}{2}\right]$ for easy ...
56 views

### How many prime ideals in $\mathbb{Z}[i]$ contain $10$?

This is an exam question I don't think I got. It asked how many prime ideals in $\mathbb{Z}[i]$ contain $10$. I know the prime ideals in $\mathbb{Z}[i]$ are principal ideals generated by primes, and ...
72 views

### Alternative methods to prove that $Z_m$ is a field under addition and multiplication $\bmod\ m$ iff $m$ is a prime

I am looking for ways to prove that $\mathbb{Z}_m=\{0,1,2,\dots,m-1\}$ is a field under addition and multiplication $\bmod\ m$ iff $m$ is a prime. I tried it this way: If $m$ is a prime $p$,it is ...
115 views

### Is it true that if $f(x)$ has a linear factor over $\mathbb{F}_p$ for every prime $p$, then $f(x)$ is reducible over $\mathbb{Q}$?

We know that $f(x)=x^4+1$ is a polynomial irreducible over $\mathbb{Q}$ but reducible over $\mathbb{F}_p$ for every prime $p$. My question is: Is it true that if $f(x)$ has a linear factor over ...
59 views

### Intersection of splitting fields of two polynomials

Let $f,g\in \mathbb{Z}[x]$, $(f,g)=h\in \mathbb{Z}[x]$ (if this is not true, can we construct $h$ in another way?). Let $K_1,K_2$ be the splitting fields of $f,g$ over $\mathbb{Q}$ respectively. Is ...
37 views

### Doubt with the ray class group

Good evening! I know this is a silly question, but I really don't realize the fact that the ray class group modulo the trivial cycle $1$ should be the ideal class group. Here $K$ is a number field ...
63 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 ...
34 views

### $f(x)\in D[x]$ is irreducible if and only if $f(x)$ is irreducible over $F[x]$.

Let $D$ be a principal ideal domain and $F$ be its quotient field. Prove that $f(x)\in D[x]$ is irreducible if and only if $f(x)$ is irreducible over $F[x]$. I only obtained the proof for ...
80 views

### Quadric equation in $\mathbb{Z}/n\mathbb{Z}$?

I would like to ask for some help about the following problem. Given is a polynomial $f(x)=ax^{2}+bx+c$ in $\mathbb{Z}/n\mathbb{Z}$, we know that this quadric equation $f(x)=0$ has exactly 8 ...
55 views

### Equality of discriminants of integral bases (statement in Ireland and Rosen, A Classical Introduction to Modern Number Theory)

I'm doing independent study and need assistance. This is taken from Ireland and Rosen's A Classical Introduction to Modern Number Theory, Chapter 12. Let F/Q be an algebraic number field, D the ring ...
46 views

### Peculiarities of an extended integer ring $\mathbb{Z}[i C]$

For an extended integer ring consisting of $\mathbb{Z}[i C] = \{ x + iC y \mid x,y \in \mathbb{Z} \}$, here $C$ is a real constant (I guess it being complex would change nothing), what real numbers ...
72 views

### Status of a question from Freeman Dyson's 1972 article

In a famous article, Freeman Dyson mentions an interesting relationship between the $\tau$ functions of number theory and the dimensions of finite-dimensional simple Lie algebras (section 2). He ...
44 views

### Find the cardinality of the set Hom$(\mathbb{Z}_{n_1}\oplus \cdots \oplus \mathbb{Z}_{n_r}, \mathbb{Z}_{m_1}\oplus \cdots \oplus \mathbb{Z}_{m_s})$?

I know that the number of group homomorphism between $\mathbb{Z}_n$ to $\mathbb{Z}_m$ is $\gcd(n, m)$. With some other relevant information like Aut$(\mathbb{Z}_n)$ is isomorphic to $U(n)$ (viz the ...
139 views

### Ring of algebraic integers of $\mathbb Q(\sqrt{-3})$ is Euclidean.

$A$ is the ring of integers generated by the cube roots of unity. How will we prove that it is an Euclidean ring? We have following results and theorem: 1) Any integral quadratic extension $A$ of ...
58 views

### Let $\pi$ denote a prime element in $\mathbb Z[i], \pi \notin \mathbb Z, i \mathbb Z$. Prove that $N(\pi)=2$ or $N(\pi)=p$, $p \equiv 1 \pmod 4$

Let $\pi$ denote a prime element in $\mathbb Z[i], \pi \notin \mathbb Z, i \mathbb Z$. Prove that $N(\pi)=2$ or $N(\pi)=p$, $p \equiv 1 \pmod 4, p$ is a prime. I know that $\pi$ is prime in ...
122 views

### Conjecture similar to Fermat's Theorem.

I was wondering about a problem which i could reduce to asking the following Does there exist a set $a,b,c$ of prime numbers such that $$a^a+b^b=c^c$$ Is it really a tough problem or do you think ...
Let $x_i,a_i\!\in\!\mathbb{Z}$. The following procedure solves a system of congruences $$x \equiv x_i\pmod{a_i}\;\;\text{ for }i\!=\!1,\ldots,n$$ when $a_i$ are pairwise coprime. Assume that ...
Let's define $\sigma(n)$ as the sum of the digits of the integer $n$ modulo $9$, having posed that $\sigma(9) = 9$. Now consider 2 number $a$ and $b$ in the set $\{1, \cdots, 9\}$. Which is the value ...