Ideals in a Quadratic Number Fields

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?

Solutions Collecting From Web of "Ideals in a Quadratic Number Fields"

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.

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