I am a high school student who, after some tinkering, came across the Abel–Ruffini theorem. I then learned about its explanation through Galois theory, particularly the result that a polynomial is solvable by radicals if and only if its Galois group is solvable. This leads to a confusion. The...

So far I have found that 2 and 7 are such primes. However, due to the exponential form, I can't compute very far for more examples (currently, get stuck at 19). More specifically, I want to know whether there exists a finite or infinite amount of these primes. I am not well versed in number...

On the IMO 1985 Longlist problem 11, it is asked to prove that $$\frac{b^{n-1}a(a + b)(a + 2b) \cdots (a + (n - 1)b)}{n!}$$ is an integer, where $a$, $b$, $n$ are integers, and $n>0$. The expression resembles a binomial coefficient and seems to have some combinatorial meaning. What would that be?

I am self studying algebraic geometry from the textbook of Daniel Perrin (Algebraic Geometry: An Introduction). On page 44 is the Proposition 3.5 which I am unable to prove and need help with. Proposition 3.5. Let $(X,O_X)$ and $(Y,O_Y)$ be two affine algebraic sets equipped with the affine...

I have a question in the proof ofCorollory $4.4$ of Daniel Perrin's Algebraic Geometry on Page $45$. Corollary $4.4$ A non empty algebraic variety can be uniquely written as a finite union of irreducible closed sets which do not contain each other. Proof:By quasi-compactness, we can write $X=...

I have 2 question in the proof of Proposition $4.6$ of algebraic geometry by Daniel Perrin.( Page $46$). Statement of Proposition $4.6$: Let $X$ be an algebraic variety and let $Y$ be a closed set in $X$. We define a sheaf of rings $O_Y$ of $Y$ by setting $O_Y(V)= ${$f:V\to k| \forall x\in V...

According to the book Number Theory II: Iwasawa Theory and Automorphic Forms by Nobushige Kurokawa, Masato Kurihara and Takeshi Saito, one way to prove the functional equation of the Dedekind eta function $\eta(-\frac{1}{z})=\sqrt{-iz}\eta(z)$ is as follows: Consider this :...

Following is an attempt at a proof (as an exercise) that the average of primes on an interval $[1,n]$ is asymptotically equal to the midpoint of the interval. While I think the idea is correct (see data below), there may be a better way, and notational improvements welcome. To show:...

Let $a, b$ be positive integers. In the Cartesian coordinate plane, consider the rectangular region $S$ (including the boundary) enclosed by the points $(1,1)$, $(1,b)$, $(a,1)$, and $(a,b)$. The line $$ l: (2a+1)y - (2b+1)x = 0 $$ divides $S$. Let $f(a,b)$ be the absolute value of the...

Let $\lambda(n) = (-1)^{\Omega(n)}$ be the Liouville function and $$ C(x) = \sum_{n \le x} \frac{\lambda(n)}{n}. $$ By Turán's classical approach, $C(x)$ is controlled by $$ \sum_{n=1}^\infty \frac{\lambda(n)}{n^{s+1}} = \frac{\zeta(2s+2)}{\zeta(s+1)}, $$ and each nontrivial zero $\rho = \beta +...

This question was asked in my assignment and I am stuck on it. Question:If $A$ is infinite integral domain , $F,G\in A[X_i|i\in I]$, $F\neq 0$ and if $A^I \setminus V_A(F)\subseteq V_A(G)$,then show that $G=0$. Attempt: Let $D(F)= A^I \setminus V_A(F)=${$a\in A^I : F(a)\neq0$} . Given...

I realize this may be an unusual request, but I am trying to find out whether this style of studying mathematics already exists, or whether there are resources that come close to it. I am not looking for a standard textbook on Galois theory, nor for a historical overview followed by the modern...

My name is Kadin Shah, and I am a rising senior at Arizona State University studying Applied Mathematics with an emphasis in mechanical engineering. I'm interested in using creative problem-solving solutions in a collaborative setting with engineers at a large company such as a National...

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 had currently passed 10th grade. I want to go in pure mathematics and research and publish paper in that field (especially Number Theory) please tell me roadmap and the books to follow one after one. Please Guide me.

For even number $n$, if $n$ is not a totient (i.e. not in the range of Euler totient function), then $n$ is not in OEIS sequence A252502, but if $n$ is a totient, must be $n$ in A252502? For odd number $n$, if $n-1$ is not a totient (i.e. not in the range of Euler totient function), then $n$ is...

Erdős Problem 252:(https://www.erdosproblems.com/252) Let $k\geqslant 1$ and $\sigma_{k}(n):= \sum_{d|n} d^{k}$. Is $\displaystyle \sum_{n=1}^{\infty} \frac{\sigma_{k}(n)}{n!}$ irrational? Currently this problem is open for $k \geqslant 5$. I am considering a different version of this problem...

I know that Schur and Murty proved that an Euclidean proof (hence a "non-analytic" proof) for the existence of infinite primes $p \equiv \ell \mod q$ with $q$ and $\ell$ coprime can be given if and only if $\ell^2 \equiv 1 \mod q$. Are there any cases where $\ell^2 \not\equiv 1 \mod q$, but we...

In this related question, @Gerry Myerson asked if the conjecture fails for composites. Upon examining for composites, I found a pattern for module $n^2$ instead of $n^3$ in the linked original question. I have verified these conjectures for $n \le 5 \times 10^6$. Can they be proved or disproved....