If $R$ is a \index{commutative ring}, then $\polyring{R}{X,Y,Z}$
is a multivariate polynomial ring with indeterminates $X$, $Y$,
and $Z$, and coefficients in $R$. If $R$ is a moreover an integral
- domain, then its fraction field is $\Frac{R}$.
+ domain, then its fraction field is $\Frac{R}$. If $x,y,z \in R$,
+ then $\ideal{\set{x,y,z}}$ is the ideal generated by $\set{x,y,z}$,
+ which is defined to be the smallest ideal in $R$ containing that set.
\end{section}
\begin{section}{Algorithm}
\begin{section}{Common}
The function $f$ applied to $x$ is $f\of{x}$. We can group terms
like $a + \qty{b - c}$ or $a + \qty{b - \sqty{c - d}}$. Here's a
- set $\set{1,2,3} = \setc{n \in \Nn[1]}{ n \le 3 }$. Here's a pair
- of things $\pair{1}{2}$ or a triple of them $\triple{1}{2}{3}$,
- and the factorial of the number $10$ is $\factorial{10}$.
+ set $\set{1,2,3} = \setc{n \in \Nn[1]}{ n \le 3 }$. The tuples go
+ up to seven, for now:
+ %
+ \begin{itemize}
+ \begin{item}
+ Pair: $\pair{1}{2}$,
+ \end{item}
+ \begin{item}
+ Triple: $\triple{1}{2}{3}$,
+ \end{item}
+ \begin{item}
+ Quadruple: $\quadruple{1}{2}{3}{4}$,
+ \end{item}
+ \begin{item}
+ Qintuple: $\quintuple{1}{2}{3}{4}{5}$,
+ \end{item}
+ \begin{item}
+ Sextuple: $\sextuple{1}{2}{3}{4}{5}{6}$,
+ \end{item}
+ \begin{item}
+ Septuple: $\septuple{1}{2}{3}{4}{5}{6}{7}$.
+ \end{item}
+ \end{itemize}
+ %
+ The factorial of the number $10$ is $\factorial{10}$.
The Cartesian product of two sets $A$ and $B$ is
$\cartprod{A}{B}$; if we take the product with $C$ as well, then