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Results tagged with number-fields
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user 2481
7
votes
Accepted
Number of imaginary quadratic field with its ideal class group has $\Bbb{Z}/2\Bbb{Z}$ as 2 part
Are there infinitely many $D$ such that $Cl_K[2] \cong \mathbb{Z} / 2\mathbb{Z}$?
It's well-known (and straighforward to show) that $Cl_K[2]$ has order $2^{r-1}$ where $r$ is the number of prime fac …
- 37k
6
votes
Class number of imaginary quadratic fields
The condition shouldn't be "$n$ is prime" but "$n$ is either 1, 2, or a prime congruent to 3 mod 4". For instance $\mathbb{Q}(-5)$ has class number 2.
The more general statement that the 2-torsion sub …
- 37k
12
votes
Sign and coefficients of fundamental unit of quadratic field
This might be useful:
Stevenhagen, Peter, The number of real quadratic fields having units of negative norm, Exp. Math. 2, No. 2, 121-136 (1993). ZBL0792.11041.
As Stevenhagen explains, if the discrim …
- 37k
4
votes
Accepted
Integrality of Atkin-Lehner operator for $\Gamma_1(N)$
Theorem. Let $\ell$ be prime, and $Q, R \ge 1$ such that $(\ell, Q, R)$ are pairwise coprime. Let $N = QR$ and for simplicity assume $N \ge 4$. Then $W_Q$ preserves $M_k(\Gamma_1(N), \mathbf{Z}[1/N, \ …
- 37k
3
votes
Accepted
Definability of orderings on a formally real number field
(Originally a comment, reposted as an answer:)
Choose a primitive element $\alpha$ of F (i.e. such that $F=\mathbf{Q}(\alpha)$). Let $f$ be its minimal polynomial. Then the data of a field ordering o …
- 37k
29
votes
Accepted
Computing (on a computer) higher ramification groups and/or conductors of representations.
You can also compute some higher ramification groups in Sage. At the moment it gives lower numbering, not upper numbering, but here it is anyway:
sage: Qx.<x> = PolynomialRing(QQ)
sage: g=x^8 + 20*x …
- 37k