Let $p$ be an odd prime and let $n \ge 3$. I have been considering the following family of polynomials. Set $$ r_n:=\frac{1}{2\sqrt{n-1}}, $$ and define $$ B(p,n) := r_n+\frac{p+1}{r_n}+\frac{2p}{r_n^{n-1}}. $$ Let $$ A(p,n) := 2+2p\left( \left\lfloor \frac{B(p,n)-2}{2p} \right\rfloor+1 \right)....

I am doing my MSc project on "p-adic numbers". In my first chapter of the project, I want to include the construction of $\mathbb{Z}_p$ with both algebraic and analytic approach. But I'm not able to find any detailed explanation of analytic approach i.e.,...

I extend $\mathbb Q$ to first adjoin the roots of $x^2-1=9$ and then the roots of $(x^2-1)^2-1=9$, i.e. $x^4-2x^2-9=0$. In the second extension, I have to determine whether the prime ideal $(3)$ ramifies. I use the Newton Polygon to only deduce that $1$ determines the ramification index, which...

I am looking for a topological or geometric interpretation of the Albert–Brauer–Hasse–Noether (ABHN) theorem, which establishes the local-global principle for central simple algebras over a global field $K$. The classical exact sequence is given by: $$0\rightarrow \text{Br}(K)\rightarrow...

I am trying to prove that the prime ideals above 5 in $K=\mathbb Q(\sqrt{359})$ are non-principal. I calculated the splitting to be $5O_K = (5,\sqrt{359} + 2)(5,\sqrt{359} + 3)$. If either of the ideals were principal their generator would have norm $\pm 5$. Showing that $a^2-359b^2 = -5$ has no...

It is well known that for a finite field $K$ of characteristic $p$ the ring of $n$-truncated Witt vectors $W_n(K)$ is used to classify field extensions of $K$ of degree $p^n$; for details see e.g. Bosch's Algebra, chapter 4.10 on general Kummer theory. Basically the upshot is, cyclic subgroups...

Let $G$ be a finite group, let $k$ be a field of characteristic $p$, and let $S_p$ be a Sylow $p$-subgroup of $G$. It is a well-known fact from modular representation theory that every irreducible representation factors through any normal $p$-subgroup of $G$. In particular, it factors through...

Let $A$ be a (associative with unit) $\mathbb{Q}_p$- and $\mathbb{Q}_q$-algebra where $p,q \in \mathbb{P}\cup \{\infty\}$. Does it follow that $p=q$? If $A$ would be a Hausdorff locally compact skew-field and topological $\mathbb{Q}_p$- and $\mathbb{Q}_q$-algebra then it would be necessarily...

This question is based on Chapter 6 of Field Arithmetic by Fried and Jarden. Let $R$ be a Dedekind domain with a quotient field $K$. Let $L/K$ be a Galois extension and let $S$ be the integral closure of $R$ in $L$. For a given $z\in S$ we have that $f$ is its irreducible polynomial in $K$. They...

I'm following these notes https://websites.math.leidenuniv.nl/algebra/localfields.pdf and i'm struggling with exercise $3.10$ regarding the valuation theory proof of Kummer-Dedekind. The statement of the problem is as follows: Let $L/K$ be an extension of number fields and $\alpha \in...

We can think of a modular form (say of weight $k$ and level $1$ for simplicity) as a holomorphic function on the upper-half plane $f:\mathbb{H} \longrightarrow \mathbb{C}$ satisfying $$ f\left( \frac{a\tau + b}{c\tau + d} \right) = (c \tau+d)^k f(\tau) $$ for all $\tau \in \mathbb{H}$, and which...

This question follows from Lemma 6.1.1 from CH 6.1 of Field Arithmetic by Fried and Jarden. It is the subsection on Decomposition groups.' The following paragraph sets up the notation used. In the construction of Decomposition groups the chapter starts by defining $R$ to be an integrally closed...

I am preparing a presentation and would appreciate feedback on the following explanation connecting the multiplicative group with Lubin–Tate theory. Let $K$ be a field and consider elements $x,y\in K^\times$ near the identity $1$. Write $$ x=1+X,\qquad y=1+Y. $$ Then the group law on $K^\times$...

The trace form $(a,b)\mapsto \mathrm{tr}(ab)$ is easily motivated as a choice of bilinear form on a number field $K$ by noting that it agrees with the (standard real) dot product on $\mathbb{R}^{r_1}×\mathbb{C}^{r_2}$ as restricted to the Minkowski embedding of $K$ (the one that sends a number...

Let $a$ be a positive integer, not a cube, so that $\alpha=\sqrt[3]a$ is irrational, and write $$R={\mathbb Z}[\alpha]=\{\,x+y\alpha+z\alpha^2\ |\ x,y,z\in\mathbb{Z}\,\}\ .$$ Let $\beta=x+y\alpha+z\alpha^2$ be a unit in $R$ with norm (product of conjugates) equal to $1$ (and not $-1$). Then...

Tate's $p$-adic uniformisation theorem for elliptic curves goes as follows: Let $K$ be a $p$-adic field, let $E/K$ be an elliptic curve with $v_K(j) \ge 0$, and let $\gamma(E/K)=-c_4/c_6 \in K^{\times}/(K^{\times})^2$. a) There is a unique $q \in K^{\times}$ with $|q|<1$ such that $E$ is...