On Grunwald-Hasse-Wang's theorem

The Theorem of Grunwald -

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...

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