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I think the best way for one to become convinced that class numbers of real quadratic fields tend to be small, is to look at the continued fraction expansion of $\sqrt{D}$. The period length of the c …

Probably, such a construction does not exist. Consider the following case: Let $F=\mathbb{Q}$, $p, q, r, s$ be four distinct positive rational primes, $K_1=\mathbb{Q}(\sqrt{pq})$, $K_2=\mathbb{Q}(\sq …

Yes. There exist many such curves: For any rational monic polynomial without multiple roots if $x+iy$ is a root, then $x-iy$ is a root as well. So if your curve $C$ satisfies $y\not =0,\\ (x,y)\in C …

I will begin by saying that $k=3$ might be a very specific case and this question is useless. Even if that is the case, I would like to know... The sum of squares function $r_k(n)$ is very famous. It …

I'm not sure about this, so please correct me (or just downvote?). Finding fields of a given discriminant has two main algorithms, depending on the context. If you want to find all fields up to a cer …

Possible Duplicate: Primes P such that ((P-1)/2)!=1 mod P Motivation comes from comments in this question, and it is interesting in its own right. These primes are sequence A055939 in OEIS. …

Given a number field K of dimension d over Q, and galois closure of dimension d! over Q (i.e galois group Sd), can we relate the discriminant of the galois closure to that of the discriminant of K? As …

Given a number field $K$, I am interested in the distribution of the $L^\infty$ norm of minimal polynomials (over $\mathbb{Z}$) of numbers in $K$. Also, it is interesting to restrict to numbers $\alph …

It is well known that for $K=\mathbb{Q}(\sqrt{D})$, $D < 0$, the non-maximal order of squarefree conductor $f$, relatively prime to $D$, has class number $$h_K \prod_{p|f} (p-(\frac{D}{p}))$$ What is …

Let $d$ be a positive integer greater than 2. Define an equivalence relation on monic integer polynomials of degree $d$: $f\sim g \iff f(k_1 x+k_2)=g(k_3 x+k_4)$ for some integers $k_1,...,k_4$. I …

Given a galois extension of number fields $L/K$ of even degree, set $n_0=\text{lcm} (\{[L_v:K_v] : v \in M_K \})$ ($L_v$ is any completion corresponding to a place dividing $v$). Does $2$ divide $n_0 …

Background Zev Chonoles recently asked the question "which number fields are monogenic?". The answers say that for a specific number field the question is hard. So, I thought, how about looking at al …

A recent article of Bhargava and Shankar, "Binary quartic forms having bounded invariants, and the boundedness of the average rank of elliptic curves" (http://arxiv.org/abs/1006.1002), addresses, amon …

I was thinking about how the standard proofs that $\pi$ is irrational use variations on $sin(\pi) =0$, in contrast to Apéry's proofs that $\zeta(2),\zeta(3)$ are irrational (the first of course also p …

Given an arithmetic surface $S$, I would like to know the following properties of its first and second Chow groups $CH^1(S), CH^2(S)$: Are they finitely generated? If so, what is the rank? What is t …