I have an expression where the variables are algebraic integers: $p4 = \frac{p12 - p41 \cdot p21}{p22}$ p12 is degree 48 and p22 is most likely degree 48 too. p41 is degree 32 and …

Let $K$ be a degree $n$ extension of ${\mathbb Q}$ with ring of integers $R$. An order in $K$ is a subring with identity of $R$ which is a ${\mathbb Z}$-module of rank $n$. Quest …

I have an equation that I think should not have too many solutions, but I don't see a way to argue this. Given $a, b, c, N \in \mathbb{N}$, how many positive integer solutions $x, …

Let $K$ be a discrete valuation field with valuation $v:K\rightarrow \mathbb Z\cup {\infty}$ which is normalized by $v(\pi)=1$ for a prime element $\pi$. Let $v:\overline K\rightar …

Let me detail the title of the question. I'm trying to give students an intuition of what the class number is. Let $K=\mathbb{Q}(\sqrt{-d})$, with $d>0$ a square-free integer, be …

Given integers $a,b,c,$ and cubic form $$ f(a,b,c) = a^3 + b^3 + c^3 + a^2 b - a b^2 + 3 a^2 c - a c^2 + b^2 c - b c^2 - 4 a b c $$ $$ f(a,b,c) = \det \left( \begin{array}{ccc …

Let $P$ be a pro-p group. Assume that there is a filtration of $P$ by normal subgroups $P_i$ such that $P_0=P$ and $P_{i+1} < P_i(i\in\mathbb N)$. Let $V$ be an $l$-adic represe …

Hi, I'm trying to understand the proof of one of the properties of the Artin L-function. I have the following doubts; Why take on $f_i =|G_{P_i}: H_{P_i}I_{G,P_i}|$, $H_{P_i}I_{G …

Let V be a variety over a number field, and let K and L be two algebraically closed What is known about the points of $V(\bar K \otimes_{\bar Q} \bar L )$ ? Are there results cla …

Dear MO, Let $K/\mathbb{Q}_2$ be a finite extension, and let $E/K$ be an elliptic curve with good ordinary reduction, and such that $\mathbb{Q}_2(j(E))=K$. Let $\rho:\operatorname …

Let $p \equiv 5 \pmod{8}, q \equiv 7 \pmod{8}$ be primes and $N = pq$. I want to show that the class number $n$ of $\mathbb{Q}(\sqrt{-N})$ satisfies $n \equiv 2 \pmod{4}$ if $\lef …

The p-adic Weiestrass lemma asserts that a power series $f(z)$ with coefficients in the ring of integers of a local field can be factored as $π^n·u(z)·p(z)$ where u(z) is a unit in …

Is there some irreducible $F \in \mathbb{Z}[x]$ such that $\mathbb{Z}[x]/(F)$ has no principal maximal ideal? Equivalently, is it possible that the $1$-dimensional integral domain …

Let $K$ be a local field (locally compact topological field) of characteristic zero. Is it true that $K$ is isomorphic to the completion of a number field under some valuations? I …

On page 156 of Milne's Class field theory notes available online here, he claims that the Hilbert class field of $K = \mathbb Q(\sqrt{-6})$ is the splitting field of $x^2+3$ but I …