All Questions
10
questions
1
vote
1
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118
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Prove that $\sqrt{-5}$ is a prime in the ring $R=ℤ[\sqrt{-5}]$.
If $R=ℤ[\sqrt{-5}]$ is a ring but not a UFD, prove that the irreducible element $\sqrt{-5}$ is a prime.
This is what I have so far.
Proof: Let $R=ℤ[\sqrt{-5}]$ be a ring but not a UFD. Since $\sqrt{-5}...
3
votes
0
answers
33
views
least upper bounds that are coprime
Given $n$ natural numbers $p_1$, $p_2$, ... $p_n$ find numbers $q_1$, $q_2$, ... $q_n$ that are pairwise coprimes such that $p_i$ ≤ $q_i$ and such that $\prod_{i=1..n} q_i$ is smallest possible.
I ...
- 31
1
vote
1
answer
106
views
Infinite primes proof based on natural logarithm
I'm trying to understand the proof outlined in this question. There are related questions, but those concern different parts of the proof.
For completeness, here it is:
I have trouble understanding ...
- 404
1
vote
1
answer
33
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Why is it that the degree of a subextension of $K(a^{1/p})/K$ must have a degree dividing $p$?
$p$ is a prime
$K$ is of characteristic not $p$
$a∈K$
$a^{1/p}∉K$
$a$ is not a root of unity
"Then if we pick an element $b∈K(a^{1/p})$, $b∉K$, then $K(b)/K$ is a non-trivial subextension, thus of ...
- 5,166
2
votes
1
answer
162
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Factorization in the Sense of Primes [closed]
Are there types prime factorizations in the sense of primes other than the Fundamental Theorem of Arithmetic (and its generalization to euclidean/principle rings), primary decomposition and Dedekind ...
- 1,457
1
vote
1
answer
40
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Find $x$ such that $[x] \neq [0]$, $[x]\in\mathbb{Z}_n$, but $[x]^2=0$.
Find $x$ such that $[x] \neq [0]$, $[x]\in\mathbb{Z}_n$, but $[x]^2=0$. (Here $[x]$ denotes the equivalence class of $x$).
My goal here is to express $x$ in terms of $p$, a prime, and $m$, a natural ...
user322548
0
votes
1
answer
375
views
Finding the $18$th cyclotomic polynomial $\phi_{18}(X))$.
I know that for an $n$th cyclotomic polynomial $\phi_n(X)$ the following equations hold:
$x^n-1=\prod_{n_1|n} \phi_{n_1}(X)$
For $n=p$ prime, $\phi_p(X)=X^{p-1}+...+X+1$
So I used the following ...
- 2,179
0
votes
1
answer
171
views
Show that every prime factor of $4t^2 + 1$ is equivalent to 1 modulo 4
Show every prime factor of $4t^2 + 1$ is equivalent to 1 modulo 4
My working so far:
I want to use the first Nebensatz, so given q is a prime factor I want to show $(-1/q)=(-1)^{(q-1)/2}=1$ as this ...
- 621
2
votes
0
answers
56
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If there is an integer $n$ such that $n^2\equiv3\pmod p$, where $p$ is prime, prove there are integers $a$ and $b$ such that $|a^2-3b^2|=p$
If there is an integer $n$ such that $n^2\equiv3\pmod p$, where $p$ is prime, prove there are integers $a$ and $b$ such that $|a^2-3b^2|=p$.
So $n^2-3 = pm$ for some integer $m$, and I know $|a^2 - ...
- 445
4
votes
1
answer
2k
views
How can I find decompositions in $\mathbb{Z}[\sqrt{d}]$?
Decompositions in $\mathbb{Z}$
In $\mathbb{Z}$ you can find a decomposition of any element $n \in \mathbb{Z}$ by factorization such that
$$n = \prod_{p \in \mathbb{P}} p^{v_p(n)}$$
So for a ...
- 9,891