Tagged Questions
4
votes
3answers
133 views
Show that there exists $f ∈ \mathbb{Z}$ such that $f^2 + f +1 ≡ 0 \pmod p$.
Let $p ≡ 1 \pmod 3$ be a prime. Show that there exists $f \in \mathbb{Z}$ such that $f^2 + f +1 \equiv 0 \pmod p$.
I know the first few primes of this form are: $7,13,19$
So for example $p=7$ we ...
1
vote
0answers
45 views
Galois invariant of Tate twists
let $k$ be the maximal extension of $\mathbb{Q}$ unramified outside a set $T$ of primes in $\mathbb{Z}$.
Take a $p\in T$ and set $G=Gal(k/\mathbb{Q})$. I would like to now if there is a classical ...
0
votes
1answer
93 views
Ideals as a product of prime ideals
Suppose we are working in $\Bbb{Q}(\sqrt{-41})$. Given a ideal, for example $(2-\sqrt{-41})$ (we especially work in $\Bbb{Z}[\omega_{-41}]$). We know that this is a Dedekind ring, thus we have unique ...
2
votes
1answer
105 views
Can we descend field extensions of prime degree of number fields to number fields of the same degree
Let $K$ be a number field and let $p$ be a prime number.
Let $L$ be a degree $p$ field extension of $K$.
Does there exist a degree $p$ field extension $M$ of $\mathbf{Q}$ such that ...
4
votes
1answer
132 views
Problems about consecutive semiprimes
I was playing around with semi-prime numbers and I made two conjectures, which are:
Given any integer $a$, at least one of $a,(a+1),(a+2)$ or $(a+3)$ is not semi-prime.
There are infinitely many ...
19
votes
1answer
650 views
Are Primes a Self-Fulfilling Prophecy?
Assume the following process:
Let's start with the set of primes $\{p_k\}$
Then we use the Euler product being equivalent to Riemann's Zeta function
$$
\prod_{p \text{ prime}} \frac{1}{1-p^{-s}} = ...
3
votes
2answers
193 views
Split prime in $\mathbb{Z}[\sqrt{14}]$
I have this assertion: if $p$ is a prime such that $p\equiv 11 \pmod{56}$, then $p$ splits in $\mathbb{Z}[\sqrt{14}]$ (the discriminant of $\mathbb{Z}[\sqrt{14}]$ is $56$.)
Why? Does $p\equiv ...
2
votes
1answer
153 views
If a prime with prime norm is a split prime, in the number ring PID
If a prime with prime norm is a split prime , in an number ring PID?
Example: $5-\sqrt{14}$ in $\mathbb{Z}[\sqrt{14}]$ has norm $11$, it is a split prime in $\mathbb{Z}[\sqrt{14}]$? Why?
Thanks
2
votes
1answer
112 views
Find a finite extension of $\mathbb{Q}$ in which all primes split
Dear all, I would be grateful if someone could provide a solution to the following problem (using decomposition and inertia groups):
Find a finite extension of $\mathbb{Q}$ in which all primes split.
...
1
vote
1answer
212 views
Invertibility of prime ideals in a number ring lying over prime numbers
I have trouble understanding an argument in the proof of the Kummer-Dedekind theorem.
I am referring to a proof given in Peter Stevenhagen's notes.
http://websites.math.leidenuniv.nl/algebra/ant.pdf
...
9
votes
2answers
356 views
Primes dividing the values of integer polynomials
Problem: Let $n$ be an integer and $p$ a prime dividing $5(n^2-n+\frac{3}{2})^2-\frac{1}{4}$. Prove that $p \equiv 1 \pmod{10}$.
The polynomial can be re-written as ...
