The Hasse norm theorem mod squares

The Hasse–Minkowski Theorem

The Hasse-Minkowski Theorem provides a characterization of the rational quadratic forms. What follows is a proof of the Hasse-Minkowski Theorem paraphrased from the book, Number Theory by Z.I. Borevich and I.R. Shafarevich [1]. Throughout this paper, some familiarity with the p-adic numbers and the Hilbert symbol is assumed and some basic facts about quadratic forms are stated without proof. Al...

The Proportion of Failures of the Hasse Norm Principle

For any number field we calculate the exact proportion of rational numbers which are everywhere locally a norm but not globally a norm from the number field.

Some Remarks on the Hasse-Arf Theorem

We give a very simple proof of Hasse-Arf theorem in the particular case where the extension is Galois with an elementary-abelian Galois group of exponent p. It just uses the transitivity of different exponents and Hilbert’s different formula. Let E/F be a finite Galois extension with Galois group G = Gal(E/F ). Let P be a place of F and let Q be a place of E lying above P . We assume that the c...

Davenport-Hasse theorem and cyclotomic association schemes

Definition. Let q be a prime power and e be a divisor of q − 1. Fix a generator α of the multiplicative group of GF (q). Then 〈α〉 is a subgroup of index e and its cosets are 〈α〉α, i = 0, . . . , e− 1. Define R0 = {(x, x)|x ∈ GF (q)} Ri = {(x, y)|x, y ∈ GF (q), x− y ∈ 〈αe〉αi−1}, (1 ≤ i ≤ e) R = {Ri|0 ≤ i ≤ e} Then (GF (q),R) forms an association scheme and is called the cyclotomic scheme of clas...

The Brauer-Hasse-Noether theorem in historical perspective