## Factorization problem in Cyclic cubic field

Let K/Q be a cubic number field. Assume that K/Q be Galois with class number 1. Therefore Gal(K/Q) is cyclic cubic group and OK is a PID. Let p be a rational prime, p doesn’t divide disc(K). Then p is splits completely or inert in K. Consider the unit group O∗K. Does there exists a … Read more

## extending pp-adic character of the local intertia to the absolute Galois group

Suppose I have a number field F, and a finite place v of F. Let E be finite extension of Fv. I start with a continuous morphism χ:O×Fv→E×. where OFv is the ring of integers of the completion Fv of F at v. My question is if χ can be extended to a continuous morphism … Read more

## An order in $\mathbb Q[G]$ which is a maximal $\mathbb Z_p$-order in $\mathbb Q_p[G]$ for finitely many primes $p$

Let $G$ be a finite group and $S$ a finite set of prime numbers. I know that every separable $\mathbb Q$-algebra $A$ contains a maximal $\mathbb Z$-order but I wonder if the following is true. Is there a $\mathbb Z$-order $\Lambda$ in $\mathbb Q[G]$ which contains $\mathbb Z[G]$ and satisfies the following two conditions 1) … Read more

## small factors of $p^k + 1$

I am a beginner in number theory and interested to know about what is the minimum value of $k$ such that $p^k+1$ has a small factor, by small I mean of $O(poly(log p))$. I don’t know how to start about solving this problem. Any hint or references in the form of books or papers or … Read more

## Height pairings of Heegner points of nontrivial conductor

I am studying Gross’s and Zagier’s proof of the BSD conjecture for elliptic curves of rank ≤1. Their calculation essentially boils down to the following ingredients: (1.) Finding a suitable imaginary-quadratic extension K of Q with no rank growth and calculating the Néron-Tate height pairings ⟨x,σ(x)⟩ of Heegner points x of conductor 1 with their … Read more

## Solve a3+2b3+4c3−6abc=1a^3 + 2b^3 + 4c^3 – 6abc = 1

From time to time I ask about units in Cubic fields. I noticed for Z[3√2] I get an analogue of the Pell equation: det without citing the Dirichlet unit theorem. Clearly the answers are a + b\sqrt[3]{2} + c\sqrt[3]{4} = (1 + \sqrt[3]{2} + \sqrt[3]{4})^n with n \in \mathbb{Z} (since we this number is unit … Read more

## Dirichlet series of a lattice ∑a∈Λ∗|det\sum_{a \in \Lambda^*} |\det(a)|^{-s}

For a lattice \Lambda of rank n in \mathbb{Q}^{n\times n} whose non-zero elements are inversible matrices, let Z(s,\Lambda) = \sum_{a \in \Lambda^*} |\det(a)|^{-s} I wonder if (and how to show) this Dirichlet series has a functional equation ? The motivation is that \zeta_K(s) the Dedekind zeta function of a number field is of the form … Read more

## Computation Hasse unit index for biquadratic fields

For a totally real (resp. imaginary) biquadratic number field K with quadratic subfields K1, K2 and K3, is there an explicit method to determine Hasse unit index (UK:UK1UK2UK3)? Answer AttributionSource : Link , Question Author : A. Maarefparvar , Answer Author : Community

## Counting incongruent isometric factorizations in the ring of integers of a number field with non-trivial class group

Let $H$ be a multiplicatively written commutative monoid. We use $\mathcal A(H)$ for the set of atoms of $H$ and $\pi_H$ for the canonical homomorphism $\mathscr F(\mathcal A(H)) \to H$, where $a \in H$ is called an atom if it is not a unit and doesn’t split into the product of two non-units, and \$\mathscr … Read more

## Sextic resolvent rings of quintic rings

In Higher composition laws IV: The parametrization of quintic rings M. Bhargava gave an explicit parametrization of quintic rings by quadruples of 5×5 skew-symmetric matrices. His proof hinges on establishing a previously unknown fundamental resolvent map between the triple product of the resolvent ring S and the dual of the quintic ring R. The mere … Read more