Tagged Questions

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Chebotarev density theorem for $k$-almost primes

Consider a finite Galois extension $L$ of $\mathbb Q$, of Galois group $G$. Let $k \geq 1$ be a fixed integer. Let $D$ be a subset of $G^k$ invariant by conjugation and by the natural action of the ...
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$p=4x^2+27y^2$,with $p$ a prime [closed]

p is a prime ,on what condition the Diophantine equation is solvable.what is it Linear expression ,for example ,$x^2+3y^2=p$, $p=3k+1$ ,$x^2+5y^2=p$ , $p=1,9\pmod{20}$.
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Class number of real maximal subfield of cyclotomic fields

Let $p$ be a prime number and $h_p^+$ the class number of $\mathbb{Q}(\zeta_p + \zeta_p^{-1})$. What is known about the values of $p$ for which $h_p^+ = 1$? Are there infinitely many? Finitely many? ...
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For any n and some prime p there is an elemnet in Zp* of order n [closed]

How can I prove, that for any positive integer $n>0$ there is a prime $p$, such that the multiplicative group of the residue ring $Z_p^*$ contains an element $a$ of order $n$? No ideas at all...
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Non-trivial facts about primes coming out of Algebraic Number Theory [closed]

What can be gleaned about primes from Algebraic Number Theory? I know this is too vague. What I mean is the following: Are there several examples where Algebraic Number Theory helps to solve ...
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Need there be infinitely many Gaussian primes along lines that contain at least one?

Greetings from EuroCG 2012, from which I post via iPod, so apologies for lack of problem motivation, background research and mathematical formatting. Question:Suppose L is a horizontal or vertical ...
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Splitting of primes in cubic fields with limited ramifications.

Let $\mathbb{F}$ be a cubic field, i.e, $\mathbb{F} = \mathbb{Q}(\alpha)$ where $\alpha$ is a root of a cubic irreducible polynomial over $\mathbb{Q}$, satisfying $disc(\mathbb{F}/\mathbb{Q})$ is a ...
For small values of $n$ and $(a, n) = 1$ it is sometimes possible to give an elementary proof that there are infinitely many primes congruent to $a \bmod n$ along the lines of Euclid's classic proof ...