Let $F/K$ be a finite field extension, $G=\text{Aut}_K(F)$ be the group of automorphisms of $F$ that fix elements of $K$, and $F^G$ be the fixed field of $G$. We then have $$[F:F^G]\le |G|.$$ This is proven in Hungerford Chapter V, Lemma 2.9. Hungerford used this lemma to prove "$F^G=K$ iff...

While creating quantitative aptitude problems for management entrance exam preparation, I noticed that the Euclidean algorithm is almost always preferred over prime factorization for computing the greatest common divisor.

I am in the middle of a problem which needs showing that the following system of congruences has finite number of solutions. I have verified up to some extent through sage that this has only two solutions for $(q, r)$ (with $q<r$) namely $(11, 17)$ and $(23, 103)$. The system of congruences is...

Is it possible to know how many steps are left to reach 1 knowing only x? The main idea is: when we analyze numbers of the form $n = 3^x + 2^x$ (for $x \ge 1$), we can track the Collatz trajectory using algebra instead of doing it number by number. Following the rules (if it is odd, multiply by...

Fix $\frac12<\sigma<1$, and define the signed logarithmic integral $$ A_\sigma(T) \int_2^T \log |\zeta(\sigma+it)|,dt. $$ I am interested in transferring recent zero-density estimates into bounds for $A_\sigma(T)$. Applying Littlewood's lemma to $\zeta(s)$ in the rectangle $$\sigma\le...

Suppose $H$ is an algebraic group (let's say over $\mathbb{R}$ or $\mathbb{C}$). I'm interested in the space $\mathrm{Sub}_{\text{alg}}(H)$ whose elements are conjugacy classes of algebraic subgroups of $H$. Is it true that $\mathrm{Sub}_{\text{alg}}(H)$ can be realized as a standard Borel...

Consider the digit dynamical system $$ f(n)=\bigl(E(n)\,O(n)\bigr)^2, $$ where $E(n)$ and $O(n)$ denote the sums of the even and odd decimal digits of a positive integer $n$, respectively. The state of an integer may be represented by the ordered pair $$ (E(n),O(n)). $$ I plotted all attainable...

I have been analyzing the problem of integer factorization by looking at non-trivial square roots of unity modulo $n$. Starting directly from the quadratic congruence $x^2 \equiv 1^2 \pmod n$, I derived a specific parameterization that maps the problem onto a generalized Pell equation of the...

Say p is 1 mod 3. Then there's a unique real cubic field in $\mathbb{Q}(\zeta_p)$. Does $p$ divide its class number? If no, this implies that the eigenspaces of the class group of $\mathbb{Q}(\zeta_p)$ corresponding to $(p-1)/3$ and $2(p-1)/3)$ vanish.

The following functions were originally proposed in a Reddit discussion on r/googology Define $$ \operatorname{LCD}(a,b)= \begin{cases} \min\{\,d>1:\ d\mid a,\ d\mid b\,\}, & \text{if such integer divisor exists},\\ 0, & \text{otherwise}. \end{cases} $$ For each positive integer $n$, let $$...

Let $$ \gamma=\lim_{x\to\infty}\left(\sum_{n=1}^x \frac1n-\log x\right) = 0.57721... $$ be Euler’s constant, and let $$ M=\lim_{x\to\infty}\left(\sum_{p \text{ prime}}^{p\le x}\frac1p-\log\log x\right)=0.26149... $$ be Mertens’ constant. These are two examples of reciprocal sums with (iterated-)...

(Moved from previous post since the answers were not elliptic curves.) I. Question We seek to find infinitely many primitive solutions to, $$a^4+b^4+c^4 = d^4+e^4$$ where $d \color{red}{\ne} e$ using polynomial solutions or elliptic curves. The most well-known case when $d = e$ is,...

For a commutative ring $R$ and an $R$-module $M$, let $\operatorname{Gr}(n,M)$ be the Grassmann functor from the category of $R$-algebras to sets. As I understand it, it should be possible to define a subfunctor $U$ when fixing $x_1, \ldots, x_n$ elements of $M$, how does this subfunctor look...

Let all terms be positive. There are infinitely many solutions to, $$a^2+b^2 = (b+1)^2$$ $$a^3+b^3+c^3 = (c+1)^3$$ like the well-known $3^2+4^2 = 5^2$ and $3^3+4^3+5^3 = 6^3$. However, this has versions for higher degrees. For $4$th powers by Jaroslaw Wroblewski, $$178^4 + 1345^4 + 10400^4 +...

(Updated with a computer search.) I. Question We seek to find infinitely many primitive solutions to, $$a^4+b^4+c^4 = d^4+e^4$$ where $d \color{red}{\ne} e$. The most well-known case when $d = e$ is, $$a^4+b^4+(a+b)^4 = 2(a^2+ab+b^2)^2$$ where one then solves $a^2+ab+b^2 = z^k$ for $k=2$. (In...

It is known that every finitely generated projective module $M$ over a commutative ring $A$ has locally constant rank, i.e., for each $\mathfrak{p} \in \mathrm{Spec}(A)$, there are non-negative integer $r$ and an open neighborhood $U \subseteq \mathrm{Spec}(A)$ such that, for every $\mathfrak{q}...

In the recent IMO 2026 Problem 1, a blackboard game is played where two active integers $m, n > 1$ are repeatedly selected and replaced by: $$g = \gcd(m, n) \quad \text{and} \quad l' = \frac{\operatorname{lcm}(m, n)}{\gcd(m, n)}$$ It is a known result that this game must terminate in a finite...

I have been studying the following digit dynamical system. Let $E(n)$ denote the sum of the even digits of $n$, and let $O(n)$ denote the sum of the odd digits of $n$. Define $$ f(n)=\big(E(n)\,O(n)\big)^2. $$ I am interested in the dynamical system obtained by repeated iteration $$...

Let $G=\operatorname{Spec}(R)$ be a finite flat commutative group scheme over $S=\operatorname{Spec}(A)$ of characteristic $p>0$. Suppose the $p$-Lie algebra $\operatorname{Lie}(G/S)$ is locally free. I would like to know the $p$-mapping on $\operatorname{Lie}(G/S)$ coincides with differential...

for the diophantine $x_1^8+x_2^8+x_3^8+x_4^8=y_1^8+y_2^8+y_3^8+y_4^8 $ Back in 2006, Nuutti Kuosa discovered the following non-trivial integer solution $$1953^8+2012^8+3113^8+861^8=1128^8+2767^8+2557^8+2823^8$$ A system of equations was used to speed up the search....