Intereting Posts

Rellich's theorem for Sobolev space on the torus
A proof that BS(1,2) is not polycyclic
Surjective endomorphism of abelian group is isomorphism
how to find a complex integral when the singular point is on the given curve
Deriving the Normalization formula for Associated Legendre functions: Stage $3$ of $4$
Show that $\left| \sqrt2-\frac{h}{k} \right| \geq \frac{1}{4k^2},$ for any $k \in \mathbb{N}$ and $h \in \mathbb{Z}$.
On terms “Orientation” & “Oriented” in different mathematical areas?
Division Rings and Division Algebras
Is $\mathbb{R}$ a vector space over $\mathbb{C}$?
The notations change as we grow up
Probability question.
If $A$ is a subobject of $B$, and $B$ a subobject of $A$, are they isomorphic?
Variance of the number of empty cells
Definite Dilogarithm integral $\int^1_0 \frac{\operatorname{Li}_2^2(x)}{x}\, dx $
Is whether a set is closed or not a local property?

In the literature it is stated that to each quadratic irrational $\gamma=\frac{P+\sqrt{D}}{Q}$ there is a corresponding ideal $I=[|Q|/\sigma , (P+\sqrt{D})/\sigma]$, where $\sigma=1$, if $\Delta \equiv0$ mod $4$ and $\sigma=2$, otherwise.

Thus, in the case of $\frac{2+\sqrt{13}}{3}$ the associated ideal must be $I=[3/2, (2+\sqrt{13})/2]$ which makes no sense, as $N(I)=3/2$ is supposed to be a rational integer.

What am I doing wrong here?

- Algorithm for determining whether two imaginary quadratic numbers are equivalent under a modular transformation
- why does a certain formula in Lang's book on modular forms hold?
- What are the units of cyclotomic integers?
- Prove that the class number of $\mathbb{Z}$ is $1$
- On the class number of a cyclotomic number field of an odd prime order
- special values of zeta function and L-functions

- Splitting of primes in an $S_3$ extension
- Ramification index of infinite primes
- On products of ternary quadratic forms $\prod_{i=1}^3 (ax_i^2+by_i^2+cz_i^2) = ax_0^2+by_0^2+cz_0^2$
- Small integral representation as $x^2-2y^2$ in Pell's equation
- A sufficient condition for a domain to be Dedekind?
- Homogeneous forms of degree $n$ in $n$ indeterminates over $\mathbb{Z}$: which ones come from the norm of a number field?
- Discriminant of splitting field
- Representation of an algebraic number as a fraction of algebraic integers which are relatively prime to a given ideal
- Subgroup generated by $1 - \sqrt{2}$, $2 - \sqrt{3}$, $\sqrt{3} - \sqrt{2}$
- Norm of the product of two regular ideals of an order of an algebraic number field

Below is a proof of the standard equivalences between forms, ideals and numbers, excerpted from section 5.2, p. 225 of Henri Cohen’s book “A course in computational algebraic number theory”. Note that your quadratic number is not of the form specified in this equivalence, viz. $\rm\ \tau = (-b+\sqrt{D})/(2a)\:,\:$ and $\rm\: 4\:a\:|\:(D-b^2)\:,\:\:$ i.e. $\rm\ a\:|\:N(a\tau)\:,\:$ a condition equivalent to the $\rm\mathbb Z$-module $\rm\ a\:\mathbb Z + a\tau\ \mathbb Z\ $ being an ideal when $\rm\:D\:$ and $\rm\:b\:$ have the same parity, e.g. see Proposition 2.8 p.18 in Franz Lemmermeyer’s notes.

- What are the surfaces of constant Gaussian curvature $K > 0$?
- $k$ cards between the two cards of rank $k$
- Graph of continuous function has measure zero by Fubini
- proving identity for statistical distance
- If every eigenvector of $T$ is also an eigenvector of $T^{*}$ then $T$ is a normal operator
- How many different arrangements?
- Compute the period of a decimal number a priori
- Prove that $PSL(2,\mathbb{Z})$ is free product of $C_2$ and $C_3$
- Orthogonal matrix over cross product
- Probability of crossing a point in a given time window
- What is the algorithm hiding beneath the complexity in this paper?
- A question with the sequence $e_{n}=\left(1+\frac{1}{n}\right)^{n}$
- $p^a\mid f(v) \implies p^a\mid f(w)$ in $\mathbb Z$
- Why is it that a linear transformation can only preserve or reduce a vector space's dimension?
- Quotient of a local ring at a point is a finite dimensional vector space