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"...

I would like some help with this question: let $S$ be a sparse subset of $\mathbb {N }$. Let $M$ be a subset of $S$ such that if $m\in M$ and $sa=m$ with $s\in S$ implies that $s=m$ and $a=1$. Let $S(x)$ be the number of represnetations of elements of $S$ less than x. We say that $c(n)$ is the...

Let $K$ be a field, and consider a morphism of locally ringed spaces (or schemes) $f: \operatorname{Spec} K \to Y$. Let $t$ be the unique topological point of $\operatorname{Spec} K$, and suppose that $f(t) \in V$, where $V$ is an affine open subset of $Y$. My question is: Can $f$ be uniquely...

I am working through exercise 14.1.B(c) in Ravi Vakil's excellent Fundamentals of Algebraic Geometry. I'll here reproduce the statement: Exercise 14.1.B(c) - Let $\pi:X \to Y$ be a morphism of ringed spaces. If $\mathscr G$ is a locally free sheaf of rank $n$ on $Y$ and $\{U_i\}$ are...

Fix a commutative ring $k$. $\def\Spec{\operatorname{Spec}}\def\Mod{\operatorname{Mod}}\def\CAlg{\operatorname{CAlg}}\def\Ab{\operatorname{Ab}}\def\Sym{\mathcal{S}}\def\V{\mathbb{V}}\def\A{\mathbb{A}}$ Let $M$ be a $k$-module and $G=\Spec H$ be an affine group scheme. After being initially...

In standard projective (P^2), the line at infinity is defined as ([X,Y,Z]) with (Z=0). However, other authors define it as ([X,Y,Z]) with (X=0). I feel some confusion. Can it be shown that this does not depend on these choices? Is the elliptic curve group structure preserved under projective...

I am tasked to prove the formal affine line $\hat{\mathbb{G}}_a$, seen as the functor from animated rings to set $$Ani(Ring)\to Ani$$ $$R\mapsto Nil(\pi_0(R))$$ taking an animated ring to the nilradical of its underlying static ring, to be an etale stack i.e. I have to show that it satisfies...

I want to read about inverse Galois Theory with the goal of proving the Hilbert Irreducibility Theorem. I do know the basics of Algebra (Group and Ring Theory, Field and Galois Theory, a bit of Moduls). Is there any good book which is on an undergraduate level about Inverse Galois Theory?