User:Ppol10/sandbox

2nd Definition

An ideal lattice is a lattice $$\textit{(I,b)}$$, where $$\textit{I}$$ is a (fractional) $$ \textit{O} $$-ideal and $$\textit{b} : \textit{I} \times \textit{I} \longrightarrow \textbf{R}$$ is such that

$$b(\lambda x,y) = b(x,\overline{\lambda} y)$$

for all $$x, y \in \textit{I} $$ and for all $$\lambda \in \textit{O} $$.

As a $$ \mathbb{Z}$$-module, $$ \mathbb{Z}[x]/\langle f \rangle $$ is isomorphic to $$ \mathbb{Z}^n$$ regardless of the choice of $$ f$$.

For simplicity, some studies only concentrate on rings of the form $$ \mathbb{Z}[x]/\langle x^n+1 \rangle $$, as they have proved to be the most useful for practical applications.

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