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?

I'm trying to figure out some facts about the pullback of Cartier divisors, but honestly I'm having a hard time describing the pullback of the sheaf associated to divisor. Let $\phi\colon X \to Y$ be a dominant morphism of schemes. Suppose $X, Y$ are both integral, noetherian, separated. If...

It is a well known fact that the Galois group $G$ of a Galois extension $K\subseteq L$ is a profinite group, as $G$ is equal to the inverse limit of the Galois groups of the finite subextensions of $K\subseteq L$. Therefore, $G$ gets a "natural" topology that turns it into a compact group. Why...

Recently we encountered a difficult problem in computer science, but since it is very closely related to mathematics, I was unsure which board would be more appropriate. In the end I posted it here on the mathematics board. To make the problem easier to understand, I will give both a...

Is the following statement true or false?Let $\mathbb{N}$ be the set of positive integers. For any $z > 2$, there always exist $x, y < z$ such that:$$f(x) + f(y) = f(z)$$Where the arithmetic function $f(n)$ is defined as:$$f(n) = \prod_{p^k \parallel n} \left( \frac{p^{k+1}-1}{p-1} \right) =...

As I mentioned in my previous post, I have been reading (Talpo and Vistoli)'s paper Infinite root stacks and quasi-coherent sheaves on logarithmic schemes. While editing that post, a second question came up, which I thought was sufficiently independent to deserve its own post. So here is the...

The discovery by Andrew Booker of an integer solution to, $$N=x^3 + y^3 +z^3=33$$ $$8866128975287528^3 - 8778405442862239^3 -2736111468807040^3=33$$ got some press and Youtube mileage back in 2019. As mentioned in Booker's July 2019 article, for $0<N<1000$, there used to be $13$ unsolved $N$,...