Intereting Posts

Solving a literal equation containing fractions.
Questions on Symmetric group of degree 5
Show that the equation $x^2+y^2+z^2= (x-y)(y-z)(z-x)$ has infinitely many solutions in integers $x, y, z$.
Representations of a quiver and sheaves on P^1
Show that a finite group with certain automorphism is abelian
Hockey-Stick Theorem for Multinomial Coefficients
Prove that inequality is true for $x>0$: $(e^x-1)\ln(1+x) > x^2$
Let $W_1$ and $W_2$ be subspaces of a finite dimensional inner product space space. Prove that $(W_1 \cap W_2)^\perp=W_1^\perp + W_2^\perp $
Use an induction argument to prove that for any natural number $n$, the interval $(n,n+1)$ does not contain any natural number.
Order of some quotient ring of Gaussian integers
Prove that a bounded sequence has two convergent subsequences.
Polynomial fitting where polynomial must be monotonically increasing
Show that the equation $y^2 = x^3 + 7$ has no integral solutions.
Legendre transform of a norm
Are Euclid numbers squarefree?

Any Euclidean domain satisfies the division “algorithm”:

For any $x,d$ there exists $q,r$ such that $x = qd+r$ with $\sigma(r)<\sigma(d)$ or $\sigma(r)=0$

With $\sigma$ a “size function.”

- What happens if we remove the requirement that $\langle R, + \rangle$ is abelian from the definition of a ring?
- Ideal of polynomials in $k$ vanishing at a point $p$ is $(X_1 - p_1, …,X_n - p_n)$
- On the relationship between the commutators of a Lie group and its Lie algebra
- A non-UFD such that $a^2 \mid b^2$ does not lead to $a\mid b$
- Prove that prime ideals of a finite ring are maximal
- Nonabelian group of order $p^3$ and semidirect products

I’m wondering if what *I* would call an algorithm (i.e. a discrete series of steps to get to an answer) exists for division. Specifically:

Suppose +, -, and * are defined in some Euclidean Domain. Is there a mechanism to find $x/y$ (beyond brute force)?

Repeated subtraction works in the integers, but not the polynomials, and this was my first thought. (I realize that I’m ignoring the problem of how to do subtraction in the ring, which is quite similar.)

- Direct product of two nilpotent groups is nilpotent and direct product of two solvable groups is solvable
- Euler Totient Issues
- Nice examples of groups which are not obviously groups
- The relationship between inner automorphisms, commutativity, normality, and conjugacy.
- How to geometrically show that there are $4$ $S_3$ subgroups in $S_4$?
- On some propreties of orthogonal complements
- Any ring is integral over the subring of invariants under a finite group action
- Prove that $\mathbb{R^*}$, the set of all real numbers except $0$, is not a cyclic group
- How to factor the ideal $(65537)$ in $\mathbb Z$?
- How to prove that $z\gcd(a,b)=\gcd(za,zb)$

Various complexity results are known about Euclidean domains. For example, Downey and Kach: Euclidean functions of computable Euclidean domains shows that there is a computable Euclidean domain having no (finite *or* transfinitely-valued) computable Euclidean function. Further there is a computable Euclidean domain with a computable Euclidean function but whose units are noncomputable, and there is a computable Euclidean domain with neither computable units nor a computable Euclidean function. See the paper and its references for much more.

- on the boundary of analytic functions
- Integral, definite integral
- Let $S=\{1,2,3,4,\dotsc,N\}$ and $X=\{ f: S \rightarrow S \mid x < y \Rightarrow f(x)\leqslant f(y)\}$. Then $|X|$ is equal to what?
- Partial sums of $\sin(x)$
- Evaluate the series $\sum_{n = 0}^\infty \frac{1}{(2n + 1)^6}$ by examining the real Fourier series of the function $f(x) := x(\pi – |x|)$
- Height one prime ideal of arithmetical rank greater than 1
- Why $C_0^\infty$ is dense in $L^p$?
- Prove the formula $ff^{-1}(B) = B \cap f(X) \subset B$ where $f: X\to Y$
- $A_{4}$ unique subgroup of $S_4$ of order $12$
- Proof of inequality $2(\sqrt{n+1}-\sqrt{n}) < \frac{1}{\sqrt{n}} < 2(\sqrt{n} – \sqrt{n-1})$ using induction
- How to prove the formulation of mode-$n$ matricization and preclusive mode-$n$ product?
- The Identity Theorem for real analytic functions
- Why doesn't mathematical induction work backwards or with increments other than 1?
- what is product of delta function with itself?
- What happens when $\lvert\omega\rvert =1$?