## Tagged Questions

1answer
206 views

### Simplifying an algebraic integer expression

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 …
0answers
366 views

### Orders in number fields

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 …
0answers
74 views

### Bounding number of solutions to an equation:

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, …
1answer
89 views

### ramification of discrete valuation field

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 …
1answer
347 views

### how to visualize the class number of an imaginary quadratic field?

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 …
1answer
323 views

### Numbers integrally represented by a ternary cubic form

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 …
0answers
60 views

### Decompositions of representations of pro-p groups

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 …
0answers
58 views

### Artin L- Function properties

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 …
0answers
61 views

### points in $V(\bar K \otimes_{\bar Q} \bar L)$ rational over tensor product of fields

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 …
1answer
201 views

### Inertia subgroup in the ordinary reduction case when $p=2$

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 …
3answers
234 views

### Computing certain class numbers modulo 4

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 …
0answers
98 views

### P-adic Weierstrass Lemma for several variables

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 …
1answer
525 views

### Principal maximal ideals in Z[x]/(F)

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 …
1answer
190 views

### local field and number field

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 …
2answers
517 views

### Exercise in Milne’s CFT notes

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 …

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