Let $E/K$ be a finite field extension, $G=\operatorname{Aut}_K(E)$ be the group of automorphisms fixing elements of $K$, and $E^G$ be the fixed field of $E$ under $G$. We know $E/E^G/K$ and so $$[E:K]=[E:E^G][E^G:K]=|G|\,[E^G:K].$$ Additionally, $[E:K]=[E:K]_s[E:K]_i,$ where $[E:K]_s$ denote the...

A field extension $F/K$ is Galois if $F^{\operatorname{Aut}_K(F)}=K$ in Hungerford's Algebra, and it is Galois if it is normal and separable in Lang's Algebra. For the finite extension, these two definitions are the same. For infinite algebraic extensions or for transcendental extensions, which...

I notice that the recently-publicized counterexample(s) to the Jacobian Conjecture are not surjective. Is that necessarily the case? That is, (Q) If $F:\mathbb C^n \to\mathbb C^n$ is algebraic and everywhere locally injective, and also surjective, must it be injective? Maybe the relevant setting...

Let $X,Y$ be two algebraic surfaces (=smooth, proper $2$-dim schemes over fixed base field $k$) and let $E \subset X$ exceptional curve, ie $E \cong \Bbb P^1$ with self intersection $E^2=-1$. By Castelnuovo's contraction theorem $E$ can be contracted to a smooth point of a smooth surface leaving...

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}...