I started from a modular experiment for integers of the form $$ N=2k+11, $$ where I iterated $$ U_0=k+1,\qquad U_{n+1}=U_n^2-k\pmod N, $$ and called an integer $N$ captured if the orbit reached a solution of $$ x^2\equiv k\pmod N. $$ After rewriting the iteration symbolically using $$...

I'm working through Vakil and encountered the following exercise. 5.1.B. EXERCISE. Exercise 3.7.F showed that there is a bijection between irreducible closed subsets and points for affine schemes (the map sending a point p to the closed subset $\overline{\{p\}}$ is a bijection). Show that this...

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...

In the proof of Theorem 8.12 in Morandi's Field and Galois Theory, the author writes: Because $KN$ is the composite of a Galois extension of $S$ with a purely inseparable (hence normal) extension, $KN/S$ is normal. Thus, $\sigma_j(K)\subseteq KN$ by Proposition 3.28. I do not understand this...

Here is a quite interesting problem I've come up with: Let $\mathbb{N}^+ = \{1, 2, 3, \dots\}$. Does there exist a sequence of sets $A_1, A_2, \dots$ such that: $1$. $A_k \subset \mathbb{N}^+$ is nonempty and finite. $2$. $A_i \cap A_j = \varnothing$ for $i \neq j$. $3$. $\bigcup_{k=1}^{\infty}...

How to prove that for any faithfully flat quasi-compact (or, more generally, fpqc) morphisms are epimorphisms? My attempt: Let $f:X\to Y$ be faithfully flat quasi-compact (or, more generally, fpqc) and suppose that $g_1,g_2: Y\to Z$ are two morphisms such that $g_1\circ f=g_2\circ f$. Since $f$...

I am working on moments of the Riemann zeta function, and want to get a numerical lower bound for the $k$th moment of $\zeta(s)$ of the form $$\int_1^T|\zeta(1/2+it)|^{2k}dt>C(k)T(log T)^{k^2}.$$ Soundararajan (https://londmathsoc.onlinelibrary.wiley.com/doi/abs/10.1112/S0025579300011438)...

Its easy to show that there are infinite amount of two consecutive numbers $n, n+1$ such that in their prime factoring all primes are in power at least two. It is because if one have such $n, n+1$ then construct another $(2n+1)^2 - 1, (2n+1)^2$ ; we start with $(288,289)$. But are there three...

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 was going through the proof of the Abstract Primitive Element Theorem and had minor concerns about how the Galois correspondence is invoked. Abstract Primitive Element Theorem: Let $K$ be an infinite field and let $L/K$ be a finite separable extension. Then there exists $\theta\in\ L$ such...

Let $$ P(\tau)=\sum_{p\ \mathrm{prime}} q^p,\qquad q=e^{2\pi i\tau}. $$ The coefficients of $P(\tau)^2$ count ordered Goldbach representations: $$ P(\tau)^2=\sum_{n\ge0} r_G(n)q^n, $$ where $r_G(n)$ is the number of ordered pairs of primes $(p_1,p_2)$ such that $$ p_1+p_2=n. $$ This is formally...

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...

In chapter 4 of J. P. Serre's "Topics in Galois Theory", he computes the Galois groups of the splitting fields over $\mathbb Q(T)$ of a few polynomials of the form $f(X,T)=f(X)-T$. He does this by calculating their ramification type (i.e. which valuations ramify in this field extension with...

Like say 2 is 7, 4, 13, 7, 4, 13, ... Again for 3, for 4 and so ob. It always ends in this 7,4,13 loop. Now we need to prove whether it will always end in this loop or not.

I need a source for the following statement: Let $q(x,y)=ax^2+bxy+cy^2$ be a binary quadratic form with $a,b,c\in\mathbb Z$ and let $p$ be a prime with $p\not\mid 2D$, where $D=b^2-4ac$ is not a square. Then the quaternary quadratic form $q(x_1,y_1)-q(x_2,y_2)$ represents all $p$-adic integers,...

Is every automorphic L-function in the Selberg class? Is every function in the Selberg class an automorphic L-function? They both have a Riemann hypothesis associated, so are they related?

This is Exercise 11 in Section 16.1 in Dummit&Foote's Abstract Algebra. Let $V$ be an affine variety over a field $k$ and let $R = k[V]$ be its coordinate ring. Let $d_t(R)$ denote the transcendence degree of the field of fractions $k(V)$ over $k$, and let $d_p(R)$ be the Krull dimension of $R$...

The Torelli map, which maps a smooth curve to its principally polarized Jacobian, defines a morphism of algebraic stacks $$\mathscr{M}_g \longrightarrow \mathscr A_g$$ between the moduli spaces of smooth, genus $g$ curves and dimension $g$ principally polarized abelian varieties respectively....

I found the following statement (in different words with different functions) on another website: $$ \sigma(n)=2\left(n+\sum_{\substack{d|n\\d<\sqrt{n}}}d\right) -1 $$ if and only if $n=392.$ That $392$ is a solution is easy to check. Whether there are other solutions depends on "the first half"...