Intereting Posts

About irrational logarithms
Using Vieta's theorem for cubic equations to derive the cubic discriminant
Totally disconnected space that is not $T_2$
How do we know that $\mathbb{Z} + \sqrt{2}\mathbb{Z}$ isn't closed?
Let $f$ be a twice differentiable function on $\mathbb{R}$. Given that $f''(x)>0$ for all $x\in \mathbb{R}$.Then which is true?
Properties of compact set: non-empty intersection of any system of closed subsets with finite intersection property
Traversing the infinite square grid
Proof that 1-1 analytic functions have nonzero derivative
How to show equinumerosity of the powerset of $A$ and the set of functions from $A$ to $\{0,1\}$ without cardinal arithmetic?
If $f: U \rightarrow \mathbb{R}^n$ differentiable such that $|f(x)-f(y)| \geq c |x-y|$ for all $x,y \in U$, then $\det \mathbf{J}_f(x) \neq 0$
Applications of cardinal numbers
Integral $\int_0^1 \log \frac{1+ax}{1-ax}\frac{dx}{x\sqrt{1-x^2}}=\pi\arcsin a$
maximum of two uniform distributions
Proof of Cauchy Riemann Equations in Polar Coordinates
Linear isometry between $c_0$ and $c$

This is a purely recreational question — I came up with it when setting an undergraduate example sheet.

Let’s go with Wikipedia’s definition of a Euclidean domain. So an ID $R$ is a *Euclidean domain* (ED) if there’s some $\phi:R\backslash\{0\}\to\mathbf{Z}_{\geq0}$ or possibly $\mathbf{Z}_{>0}$ (I never know what $\mathbf{N}$ means, and the Wikipedia page (at the time of writing) uses $\mathbf{N}$ as the target of $\phi$, but in this

case it doesn’t matter, because I can just add one to $\phi$ if necessary) such that the usual axioms hold.

Now onto subrings of the rationals. The subrings of the rationals turn out to be in bijection with the subsets of the prime numbers. If $X$ is a set of primes, then define $\mathbf{Z}_X$ to be the rationals $a/b$ with $b$ only divisible by primes in $X$. Different sets $X$ give different subrings, and all subrings are of this form. This needs a little proof, but a little thought, or a little googling, leads you there.

- Any ring is integral over the subring of invariants under a finite group action
- Is $\mathbb{Z}$ the only totally-ordered PID that is “special”?
- Krull dimension of the injective hull of residue field
- A non-noetherian ring with all localizations noetherian
- Certain products of mostly diagonal matrices are nonzero
- $R\subseteq S$ integral extension and $S$ Noetherian implies $R$ Noetherian?

If $X$ is empty, then $\mathbf{Z}_X=\mathbf{Z}$, which is an ED: the usual $\phi$ taken is $\phi(x)=|x|$.

If $X$ is all the primes then $\mathbf{Z}_X=\mathbf{Q}$ and this is an ED too (at least according to Wikipedia — I think some sources demand that an ED is not a field, but let’s not go there); we can just let $\phi$ be constant.

If $X$ is all but one prime, say $p$, then $\mathbf{Z}_X$ is the localisation of $\mathbf{Z}$ at $(p)$, and $\phi$ can be taken to be the $p$-adic valuation (if we’re allowing $\phi$ to take the value zero, which we may as well). Note however that this is a rather different “style” of $\phi$ to the case $X$ empty: this $\phi$ is “non-archimedean” in origin, whereas in the case of $X$ empty we used an “archimedean” $\phi$. This sort of trick generalises to the case where $X$ is all but a finite set of primes — see the “Dedekind domain with only finitely many non-zero primes” example on the Wikipedia page.

Of course the question is: if $X$ is now an *arbitrary* set of primes, is $\mathbf{Z}_X$ an ED?

- Why is quadratic integer ring defined in that way?
- Kähler differential over a field
- Scheme: Countable union of affine lines
- Proof of Legendre's theorem on the ternary quadratic form
- What does the topology on $\operatorname{Spec}(R)$ tells us about $R$?
- Bounded index of nilpotency
- what numbers are integrally represented by this quartic polynomial (norm form)
- Every maximal ideal is principal. Is $R$ principal?
- Is '10' a magical number or I am missing something?
- Generalization of Dirichlet's theorem

Yes. Let $\phi(a/b) = |a|$ where $a/b$ is written in lowest terms. To see that this is a Euclidean function, let $a/b,c/d\in \mathbb{Z}_X$ be nonzero and in lowest terms and write

$$\frac{a}{b}=\frac{nd}{b}\cdot \frac{c}{d}+\frac{s}{t}$$ which means that $\phi(s/t)=\phi((a-nc)/b)\leq |a-nc|$ which for a suitable value of $n$ is less than $\phi(a/b) = |a|$.

- lim (a + b) when lim(b) does not exist?
- Integral $ \int_0^{2\pi}{ \sqrt{ 1 – \sin{ \theta } \sin{ 2\theta } + \cos{\theta}\cos{2\theta} } d\theta } $
- Sum of $\Gamma(n+a) / \Gamma(n+b)$
- Why can't prime numbers satisfy the Pythagoras Theorem? That is, why can't a set of 3 prime numbers be a Pythagorean triplet?
- Prove the fractions aren't integers
- How many total order relations on a set $A$?
- Tensor products commute with direct limits
- Probability of two people meeting during a certain time.
- Bijection between closed uncountable subset of $\Bbb R$ and $\Bbb R$.
- Tychonoff theorem (1/2)
- Is $\frac{1}{\exp(z)} – \frac{1}{\exp(\exp(z))} + \frac{1}{\exp(\exp(\exp(z)))} -\ldots$ entire?
- Intuition behind variational principle
- With infinite size, we can have $P \cdot M = M \cdot D $ (D diagonal) but where $M^{-1}$ does not exist. Can we say “P is diagonalizable”?
- If $A$ is a subobject of $B$, and $B$ a subobject of $A$, are they isomorphic?
- Splitting of the tangent bundle of a vector bundle and connections